Group analysis of general Burgers-Korteweg-de Vries equations
Stanislav Opanasenko, Alexander Bihlo, Roman O. Popovych
TL;DR
This paper performs a comprehensive group classification of variable-coefficient Burgers-Korteweg-de Vries equations, introducing new techniques for analyzing equivalence groups and Lie reductions, and classifying equations based on their symmetry properties.
Contribution
It provides the first complete group classification of (1+1)-dimensional variable-coefficient Burgers-Korteweg-de Vries equations, including new methods for analyzing equivalence groups and Lie reductions.
Findings
Classified Lie symmetries for the entire class and subclasses
Developed an original technique for classifying Lie reductions
Constructed extended generalized equivalence groups for specific subclasses
Abstract
The complete group classification problem for the class of (1+1)-dimensional th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie…
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Group analysis of general Burgers–Korteweg–de Vries equations
Stanislav Opanasenko*†‡, Alexander Bihlo†* and Roman O. Popovych*‡§*,
†* Department of Mathematics and Statistics, Memorial University of Newfoundland,
St. John’s (NL) A1C 5S7, Canada*
‡* Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01004 Kyiv, Ukraine*
§* Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czechia *
E-mail:* [email protected], [email protected], [email protected]*
The complete group classification problem for the class of -dimensional th order general variable-coefficient Burgers–Korteweg–de Vries equations is solved for arbitrary values of greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.
1 Introduction
The development of new powerful tools and techniques of group analysis of differential equations during the last decade has essentially extended the range of effectively solvable problems of this branch of mathematics. In particular, it became possible to study admissible transformations and Lie symmetries for systems of differential equations from complex classes parameterized by a few functions of several arguments.
A number of evolution equations that are important in mathematical physics are of the general form
[TABLE]
Here and in the following the integer parameter is fixed, and . We require the condition guaranteeing that equations from the class (1) are nonlinear and of genuine order . Throughout the paper we use the standard index derivative notation , , and also , , and .
The class (1) contains a number of the prominent classical models of fluid mechanics, including
[TABLE]
where ’s are constants, and the coefficient of the highest-order derivative is nonzero.
Due to the importance of equations from the class (1), there exist already a lot of papers in which particular equations of the form (1) have been considered in light of their symmetries, integrability, exact solutions, etc. Here we review only papers on admissible transformations and group classification of classes related to the class (1).
Particular subclasses of the class (1) with small values of were the subject of a number of papers published over the past twenty-five years. Thus, the equivalence groupoids of the class of variable-coefficient Burgers equations of the form with and of its subclass of equations with were constructed by Kingston and Sophocleous in [14] as sets of point transformations in pairs of equations. In fact, it was implicitly proved there that these classes are normalized in the usual sense. Later such transformations were called allowed [10, 47] or form-preserving [15], and they had preceded the notion of admissible transformations, which is of central importance in group analysis of classes of differential equations. Solving of the group classification problem for the above subclass with was initiated in [7, 46] and completed in [33, 41]. Most recently, an extended symmetry analysis of the class with considered in [14] was comprehensively carried out in [32]. The partial preliminary group classification problem for the related class of Burgers equations with sources, which are of the form , with respect to the maximal Lie symmetry group of the Burgers equation was considered in [23].
In [10, 47] allowed transformations were computed for the class of variable-coefficient KdV equations of the form with and were then used in [10] to carry out the group classification of this class; see [45] for a modern interpretation of these results. An attempt to reproduce these results for the class of variable-coefficient Burgers equations of the form with was made in [38] supposing that admissible transformations of this class are similar to admissible transformations of its third-order counterpart but in fact the structure of the corresponding equivalence groupoid is totally different from that in [10]. Transformational properties of the class of variable-coefficient KdV equations of the form
[TABLE]
were studied in [37]; see also [39]. This class coincides with the class (with the particular value ) arising in Section 9 of the present paper. The class was proved to be normalized in the usual sense and mapped by a family of its equivalence transformations, e.g., to its subclass of equations of the form with , which is also normalized in the usual sense, and solving the group classification problem in the class is equivalent to that in the subclass; cf. Proposition 7 and Section 9. Variable-coefficient generalizations of the Kawahara equation were studied in a similar way in [17, 18]. The group classification of Galilei-invariant equations of the form was carried out in [5, 8].
Unfortunately, the results obtained in many papers were incomplete or even faulty; for a partial listing of papers with such results on variable-coefficient Korteweg–de Vries equations, we refer to [37]. It is thus appropriate to solve the group classification problem for the general class (1) systematically for the first time.
More general classes of evolutions equations, which include the class (1), were also considered in the literature. Contact symmetries of -dimensional evolution equations were studied by Magadeev [24]. More specifically, Magadeev proved that a -dimensional evolution equation admits an infinite-dimensional Lie algebra of contact symmetries if and only if it is linearizable by a contact transformation. He also classified, up to contact equivalence transformations of evolution equations, realizations of finite-dimensional Lie algebras by contact vector fields with the independent variables and the dependent variable each of which is the maximal contact invariance algebras of a non-linearizable evolution equation. At the same time, evolution equations admitting these realizations as maximal contact invariance algebra were not constructed. Moreover, singling out the classification of contact symmetries of equations from a particular subclass of the entire class of -dimensional evolution equations is in general a more challenging problem than the direct classification of contact symmetries for equations from the subclass. A claim similar to the above one on results of [24] is also true for classifications of low-order evolution equations (with ) considered, e.g., in [1, 11, 12, 49].
In the present paper we solve the complete group classification problem of the class (1) for any fixed value of . An even more important result of the paper is the construction, in the course of the study of admissible transformations within subclasses of the class (1), of several examples of classes with specific properties, the existence of which was in doubt for a long time. These examples give us unexpected insights into the general theory of equivalence transformations within classes of differential equations. In particular, they justify introducing the notion of effective generalized equivalence group.
We also point out that the class of linear (in general, variable-coefficient) evolution equations is obtained from (1) by setting . This class has transformational properties different from those of the class (1), and equations with and are not related to each other by point or contact transformations. Moreover, the class of linear equations was completely classified in [21] and [30, pp. 114–118] for and in [3] for . (The last paper enhanced and extended results of [11, Section III] on and of [13] on .) This is why it is appropriate to exclude linear equations from the present consideration.
The further structure of this paper is the following. In Section 2 we present sufficient background information on point transformations in classes of differential equations and on the algebraic method of group classification. Since the class (1) is normalized in the usual sense, particular emphasis is given to describing the concept of normalization of a class of differential equations. The notion of effective generalized equivalence group is introduced there for the first time. Using known results on admissible transformations of the entire class of th order evolution equations and its wide subclasses, in Section 3 we compute the equivalence groupoids for the class (1) and its two subclasses singled out by the arbitrary-element gauges and , which are realized by families of equivalence transformations. Due to each of the above classes being normalized in the usual sense, its equivalence groupoid coincides with the subgroupoid induced by the usual equivalence group of this class. The solution of the group classification problem for the class (1) is shown to reduce to that for its gauged subclass associated with the constraint and referred to in the paper as the class (5). Moreover, this subclass turns out to be the most convenient for carrying out the group classification since it is normalized in the usual sense and maximally gauged. The consideration of alternative arbitrary-element gauges in Section 4 additionally justifies the selection of the subclass with for group classification. This also allows us to construct for the first time examples of classes with nontrivial generalized equivalence groups, where transformation components for variables locally depend on nonconstant arbitrary elements. The determining equations for Lie symmetries of equations from the class (1) are derived and preliminarily studied in Section 5. These results are used in Section 6 for analyzing properties of appropriate (for group classification) subalgebras of the projection of the equivalence algebra of the class (5) to the space of equation variables. The group classification of the gauged subclass (5) and thus the entire class (1) is completed in Section 7 via classifying the appropriate subalgebras and finding the corresponding values of the arbitrary-element tuple. In Section 8, we discuss the optimality of chosen inequivalent representatives for cases of Lie symmetry extensions. The subclass of equations from the class (1) with coefficients depending at most on is the object of the study in Sections 9. We exhaustively describe the equivalence groupoid of this subclass, gauge its arbitrary elements by equivalence transformations and single out a complete list of inequivalent Lie symmetry extensions within this subclass from that for the class (1). In fact, we begin with the wider subclass , where and may affinely depend on , since this subclass is normalized in the usual sense with respect to its nice usual equivalence group whereas the subclass , where only may affinely depend on , and the subclass give new nontrivial examples of classes normalized only in the generalized sense and only in the extended generalized sense, respectively. Similar results for the subclass of equations from the class (1) with coefficients depending at most on are obtained in Section 10. Since this subclass is not normalized in any appropriate sense, in order to describe its equivalence groupoid , it is necessary to solve a complicated classification problem for values of the arbitrary-element tuple that admit nontrivial admissible transformations, which are not generated by transformations from the corresponding usual equivalence group. It turns out that the equivalence groupoid has an interesting structure related to Lie symmetry extensions in the subclass although the subclass is far from even being semi-normalized. Lie reductions of equations from the class (1) are classified in Section 11 with respect to the usual equivalence group of this class using the fact that it is normalized in the usual sense. Other possibilities for finding exact solutions of these equations are also discussed. The conclusions of the paper are presented in the final Section 12.
2 Algebraic method of group classification
Basic notions and results underlying the algebraic method of group classification in its modern advanced form as will be used below were extensively discussed in [2, 3, 19, 34, 35, 36], to which we refer for further details. Examples of applying various versions of the algebraic method to particular group classification problems can be found in [1, 9, 10, 11, 13, 24, 27, 49]. In this section we not only review the needed part of the known theory on symmetry analysis in classes of differential equations but also present some new notions and results of this field.
Let denote a system of differential equations of the form . Here, are the independent variables, are the dependent variables, and is a tuple of differential functions in . We use the standard short-hand notation to denote the tuple of derivatives of with respect to up to order , which also includes the ’s as the derivatives of order zero. The system is parameterized by the tuple of functions , called the arbitrary elements, which runs through the solution set of an auxiliary system of differential equations and inequalities in , and, e.g., . Here, the notation encompasses the partial derivatives of the arbitrary elements up to order with respect to both and . Thus, the class of (systems of) differential equations is the parameterized family of systems ’s, such that lies in .
For the specific class of general Burgers–KdV equations (1) considered below, we have , , and, in the more traditional notation, and . The tuple of arbitrary elements is , which runs through the solution set of the auxiliary system
[TABLE]
where is a multi-index, , , and . Satisfying the auxiliary differential equations is equivalent to the fact that the arbitrary elements do not depend on derivatives of . The inequality ensures that equations from the class (1) are both nonlinear and of order .
Group classification of differential equations is based on studying how systems from a given class are mapped to each other. This study is formalized in the notion of admissible transformations, which constitute the equivalence groupoid of the class . An admissible transformation is a triple , where are arbitrary-element tuples associated with systems and from the class that are similar to each other, and is a point transformation in the space of that maps to .
A related notion of relevance in the group classification of differential equations is that of equivalence transformations. Usual equivalence transformations are point transformations in the joint space of independent variables, derivatives of up to order and arbitrary elements that are projectable to the space of for each , respect the contact structure of the th order jet space coordinatized by the -jets and map every system from the class to a system from the same class. The Lie (pseudo)group constituted by the equivalence transformations of is called the usual equivalence group of this class and denoted by . If the arbitrary elements depend at most on derivatives of up to order , then one can assume that equivalence transformations act in the space of instead of the space of .
The usual equivalence group gives rise to a subgroupoid of the equivalence groupoid since each equivalence transformation generates a family of admissible transformations parameterized by ,
[TABLE]
Here denotes the projection of the space of to the space of equation variables only, . The pushforward of by is then just the restriction of to the space of .
The usual equivalence algebra of the class is an algebra associated with the usual equivalence group and constituted by the infinitesimal generators of one-parameter groups of equivalence transformations. These infinitesimal generators are vector fields in the joint space of that are projectable to for each . Since equivalence transformations respect the contact structure on the th order jet space, the vector fields from inherit this compatibility, meaning their projections to the space of coincide with the th order prolongations of the associated projections to the space of .
In the case when the arbitrary elements ’s are functions of only, we can assume that equivalence transformations of the class are point transformations of mapping every system from the class to a system from the same class. The projectability property for equivalence transformations is neglected here. Then these equivalence transformations constitute a Lie (pseudo)group called the generalized equivalence group of the class . See the first discussion of this notion in [25, 26] with no relevant examples and the further development in [34, 36]. Often the generalized equivalence group coincides with the usual one; this situation is considered as trivial. Similar to usual equivalence transformations, each element of generates a family of admissible transformations parameterized by ,
[TABLE]
and thus the generalized equivalence group also generates a subgroupoid of the equivalence groupoid .
Definition 1**.**
We call any minimal subgroup of that generates the same subgroupoid of as the entire group does an effective generalized equivalence group of the class .
The uniqueness of an effective generalized equivalence group is obvious if the entire group is effective itself; cf. Remark 11 below. At the same time, there exist classes of differential equations, where effective generalized equivalence groups are proper subgroups of the corresponding generalized equivalence groups that are even not normal. Hence each of these effective generalized equivalence groups is not unique since it differs from some of subgroups non-identically similar to it, and all of these subgroups are also effective generalized equivalence groups of the same class. See the discussion of particular examples in Remark 28 below.
Suppose that the class possesses parameterized non-identity usual equivalence transformations and some of its arbitrary elements are constants. Then this class necessarily admits purely generalized equivalence transformations. Indeed, we can set all parameters of elements from the usual equivalence group depending on constant arbitrary elements, which gives generalized equivalence transformations. The set of such transformations is a subgroup of the generalized equivalence group . If , the usual equivalence group is an effective generalized equivalence group of the class .
The generalized equivalence algebra and an effective generalized equivalence algebra of the class are the algebras associated with the generalized equivalence group and with an effective generalized equivalence group of this class and are constituted by the infinitesimal generators of one-parameter subgroups of these groups, respectively. These infinitesimal generators are vector fields in the joint space of .
The property for equivalence transformations to be point transformations with respect to arbitrary elements can also be weakened. We formally extend the arbitrary-element tuple of the class with virtual arbitrary elements that are related to initial arbitrary elements by differential equations and thus expressed via initial arbitrary elements in a nonlocal way. Denote the reparameterized class by . Suppose that the usual (resp. generalized or effective generalized) equivalence group of induces the maximal subgroupoid of the equivalence groupoid among the classes obtained from by similar reparameterizations, and the extension of the arbitrary-element tuple for is minimal among the reparameterized classes giving the same subgroupoid of as . Then we call the group an extended equivalence group (resp. an extended generalized equivalence group) of the class .
A point symmetry transformation of a system is a point transformation in the space of that preserves the solution set of . Each point symmetry transformation of gives rise to the single admissible transformation of the class . The point symmetry transformations of the system constitute the maximal point symmetry group of this system. The common part of all is called the kernel of maximal point symmetry groups of systems from the class , . The infinitesimal counterparts of the maximal point symmetry group and the kernel , which are called the maximal Lie invariance algebra of and the kernel Lie invariance algebra of systems from , consist of the vector fields in the space of generating one-parameter subgroups of and , respectively.
Group analysis of differential equations becomes much simpler when working with infinitesimal counterparts of objects consisting of point transformations. Thus, the problem on Lie (i.e., continuous point) symmetries of a system reduces to constructing the maximal Lie invariance algebra and, therefore, is linear in contrast to the similar problem on general point symmetries. Under certain quite natural conditions on the system , the infinitesimal invariance criterion states that a vector field in the space of belongs to the maximal Lie invariance algebra if and only if the condition is identically satisfied on the manifold defined by the system and its differential consequences in the jet space . Here is the standard th order prolongation of the vector field , see [29, 30] and Section 5.
The group classification problem for the class is to list all -inequivalent values for such that the associated systems, , admit maximal Lie invariance algebras, , that are wider than the kernel Lie invariance algebra . Further taking into account additional point equivalences between obtained cases, provided such additional equivalences exist, one solves the group classification problem up to -equivalence. Restricting the consideration to Lie symmetries is important for the group classification problem to be well-posed within the framework of classes of differential equations.
When applied to systems from a class , the infinitesimal invariance criterion yields, after splitting with respect to the parametric derivatives of , a system of the determining equations for the components of Lie symmetry generators of these systems, which is in general parameterized by the arbitrary-element tuple . It is quite common that there is a subsystem of the determining equations that does not involve the tuple of arbitrary elements and hence may be integrated in a regular way. The remaining part of the determining equations that explicitly involve the arbitrary elements is referred to as the system of classifying equations. In this setting, the group classification problem reduces to the exhaustive investigation of the classifying equations. The direct integration of the classifying equations up to -equivalence is usually only possible for classes of the simplest structure, e.g., classes involving only constants or functions of a single argument as arbitrary elements, see, e.g., examples in [30]. Since most classes of interest in applications are of more complicated structure, various methods have to be used, which at least enhance the direct method [28, 42, 43].
The most advanced classification techniques rest on the study of algebras of vector fields associated with systems from the class under consideration and constitute, in total, the algebraic method of group classification. For this method to be really effective, the class has to possess certain properties, which are conveniently formulated in terms of various notions of normalization. The class of differential equations is normalized in the usual (resp. generalized, extended, extended generalized) sense if the subgroupoid induced by its usual (resp. generalized, extended, extended generalized) equivalence group coincides with the entire equivalence groupoid of .
The normalization of in the usual sense is equivalent to the following conditions. The transformational part of each admissible transformation does not depend on the fixed initial value of the arbitrary-element tuple and, therefore, is appropriate for any initial value of . Moreover, the prolongation of to the space of and the further extension to the arbitrary elements according to the relation between and gives a point transformation in the joint space of . Then and for any , and hence the group classification of the class reduces to the classification of certain -inequivalent subalgebras of the equivalence algebra or, equivalently, to the classification of certain -inequivalent subalgebras of the projection .
If the class is normalized in the generalized sense, the expression for transformational parts of admissible transformations may involve arbitrary elements but only in a quite specific way. The equivalence groupoid is partitioned into families of admissible transformations parameterized by the source arbitrary-element tuple, and the transformational parts of admissible transformations from each of these families jointly give, after the extension to the arbitrary elements according to the relation between the corresponding source and target arbitrary elements, a point transformation in the joint space of . Then and for any .
The class is called semi-normalized in the usual sense if for any there exist , and such that and . One of the transformations and can always be assumed to coincide with the identity transformation. Semi-normalization in the generalized sense is defined in a similar way. Roughly speaking, a class is semi-normalized in a certain sense if its equivalence groupoid is generated by its relevant equivalence group jointly with point symmetry groups of systems from this class. Each normalized class is semi-normalized in the same sense, and the converse is not in general true. There are also more sophisticated notions, uniform semi-normalization and weak uniform semi-normalization, which mediate the notion of normalization and semi-normalization [19, 20].
To establish the normalization properties of the class one should compute its equivalence groupoid , which is realized using the direct method. Here one fixes two arbitrary systems from the class, and , and aims to find the (nondegenerate) point transformations, : , , , , connecting them. For this, one changes the variables in the system by expressing the derivatives in terms of and derivatives of the functions and as well as by substituting and for and , respectively. The requirement that the resulting transformed system has to be satisfied identically for solutions of leads to the system of determining equations for the components of the transformation .
In the case of a single dependent variable (), all the above notions involving point transformations can be directly extended to contact transformations.
3 Equivalence groupoid
We now compute the equivalence groupoid and equivalence group of the class (1) using the direct method. Equivalence transformations will be used to find an appropriate gauged subclass of (1) that is suitable for carrying out the complete group classification. The presentation closely follows [3]. In particular, it is convenient to start with the wide superclass of general -dimensional th order () evolution equations of the form
[TABLE]
and sequentially narrowing it until the class (1) and its gauged subclasses are reached. The advantage of this method is that one can infer the normalization properties of the class (1) by keeping track of the normalization properties of the class (2) and its relevant subclasses. This not only gives restrictions on the transformational part of admissible transformations in the class (2) and its subclasses, but also leads to a more and more constrained relation between the initial and target arbitrary elements until this relation is sufficiently specified.
It was established in [24] that a contact transformation of the independent variables and the dependent variable connects two fixed equations from the class (2) if and only if its components are of the form , and provided that the usual nondegeneracy assumption and contact condition hold,
[TABLE]
The prolongation of the transformation to the first derivatives is given by
[TABLE]
if or , respectively. Such transformations prolonged to the arbitrary element according to
[TABLE]
constitute the contact usual equivalence group of the class (2) and thus this class is normalized in the usual sense with respect to contact transformations. It is also normalized in the usual sense with respect to point transformations. The point equivalence groupoid and the point usual equivalence group are singled out from their contact counterparts by the condition .
Consider the subclass of the class (2) singled out by the constraints
[TABLE]
cf. [44]. Due to the constraints , , it follows that contact admissible transformations in the subclass are induced by point admissible transformations. In other words, the contact equivalence groupoid of coincides with the first prolongation of the point equivalence groupoid of . Then the constraints and , , successively imply and for any admissible transformation in , i.e., its transformational part is of the form
[TABLE]
The prolongations of these transformations to the arbitrary element constitute the usual equivalence group of class . Therefore, the class is normalized in the usual sense.
To single out the class (1), we should set more constraints for . The complete system of these constraints is given by
[TABLE]
Then we should also reparameterize the obtained subclass, assuming as the tuple of arbitrary elements instead of . Using the direct method for computing the equivalence groupoid of the class (1), we first fix two arbitrary equations and from the class (1) and require that they are connected through a point transformation of the form (3). This particular form can be posed for admissible transformations since the class (1) is a subclass of the normalized class . It is thus necessary to re-express the jet variables in terms of . In view of (3), the expressions for the transformed total derivative operators are
[TABLE]
Substituting the expressions for the transformed values into yields an intermediate equation . Because the equations and are by assumption connected by , the equation has to be satisfied by all solutions of . We assume as the leading derivative in and substitute the expression for obtained from into . This leads to an identity, which can be split with respect to the parametric derivatives , …, . The condition that the coefficient of in has to be zero requires . Collecting the other coefficients of powers of parametric derivatives, we derive the formulas pointwise relating and with no constraints for , , and . These formulas are quite cumbersome (although obtainable using the Faà di Bruno’s formula). In addition, they are not needed at the present stage since we can first fix a suitable gauged subclass of the class (1). To do this, we only need the transformation component for the arbitrary element , which is readily obtained without using Faà di Bruno’s formula,
[TABLE]
Proposition 2**.**
The class (1) is normalized in the usual sense. Its usual equivalence group consists of the transformations in the joint space of whose -components are of the form
[TABLE]
where , , and are arbitrary smooth functions of their arguments such that .
We can now use the family of equivalence transformations parameterized by the arbitrary element , where
[TABLE]
to map the class (1) onto its subclass singled out by the constraint . Under this gauging we derive that and thus , i.e., and . The gauged subclass is still normalized in the usual sense. Moreover, due to the most prominent equations from the class (1) being of the form with , this gauge is quite natural.
Remark 3**.**
Given a class of differential equations, if a subclass is singled from it by constraints with explicit expressions for some arbitrary elements, we reparameterize this subclass using the reduced tuple of arbitrary elements, which is obtained from the complete tuple by excluding the constrained arbitrary elements. For example, under the gauge we can assume that the tuple of arbitrary elements for the corresponding subclass is .
With the restrictions on , and obtained thus far, we now complete the procedure for finding the equivalence groupoid of the subclass associated with gauge , which consists of equations of the form
[TABLE]
Transforming the equations from the gauged subclass, we find
[TABLE]
which can be written, after substituting for in view of the equation (4), as
[TABLE]
Splitting this equation with respect to yields the transformation components for the arbitrary elements. We have thus proved the following theorem.
Theorem 4**.**
The class (4) of reduced -dimensional general th order Burgers–KdV equations with is normalized in the usual sense. Its usual equivalence group is constituted by the transformations of the form
[TABLE]
where , and , , and are arbitrary smooth functions of their arguments such that .
At this stage, it is convenient to introduce one more gauge. In particular, the family of equivalence transformations parameterized by the arbitrary element with , , and maps the associated gauged subclass (4) with to the subclass of (1) with and . This gauging implies that . The corresponding gauged subclass consisting of equations of the form
[TABLE]
where and is the reduced arbitrary-element tuple, is still normalized in the usual sense. This is the gauged subclass that is appropriate for solving the complete group classification problem for the class (1). This leads to the following theorem.
Theorem 5**.**
The class (5) of reduced -dimensional general th order Burgers–KdV equations, which is singled out from the class (1) by the gauge , is normalized in the usual sense. Its usual equivalence group consists of the transformations of the form
[TABLE]
where , and , and are arbitrary smooth functions of their arguments with .
Corollary 6**.**
The equivalence algebra of the class (5) of -dimensional general th order Burgers–KdV equations is given by where , and run through the set of smooth functions of , with
[TABLE]
Proof.
Since we have already computed the usual equivalence group , the associated equivalence algebra can be obtained in a straightforward deductive fashion. In particular, is spanned by vector fields representing the infinitesimal generators of one-parameter subgroups of the usual equivalence group . Thus, successively assuming one of the parameter functions , and to depend on a continuous group parameter (in such a manner that the identical transformation corresponds to the value ), we can obtain the coefficients of the infinitesimal generators of the form by determining
[TABLE]
This results in the generating vector fields , and , which are associated to the parameter functions , and , respectively. ∎
Since the class (1) is mapped onto the class (5) by a family of equivalence transformations, and both the classes are normalized in the usual sense, the following assertion is obvious.
Proposition 7**.**
The group classification of the class (1) reduces to that of its subclass (5). More specifically, any complete list of -inequivalent Lie symmetry extensions in the class (5) is a complete list of -inequivalent Lie symmetry extensions in the class (1).
4 Alternative gauges
We show that the gauge is the best initial gauge for the class (1) and the gauge is the best for singling out a subclass in order to carry out the group classification.
An obvious choice for an alternative gauge is . It was used in [3] as the basic gauge in the course of group classification of linear equations of the form (1), for which . The -component of equivalence transformations in the class (1) is
[TABLE]
If and , then the parameters of the corresponding admissible transformations given in Proposition 2 satisfy the constraint , i.e., , where , which makes the parameterization of the usual equivalence group of the corresponding subclass more complicated than using the gauge .
Proposition 8**.**
The subclass of the class (1) singled out by the constraint is normalized in the usual sense. Its usual equivalence group is constituted by the transformations of the form
[TABLE]
where , and , , and are arbitrary smooth functions of their arguments such that , as well as if is odd and , with if is even.
In contrast to the class (4), the additional gauge slightly worsens the normalization property. It leads to the appearance of the arbitrary element in the -component of admissible transformations since then we have
[TABLE]
Denote by the arbitrary-element tuple reduced by the double gauge,
[TABLE]
Proposition 9**.**
The equivalence groupoid of the subclass of the class (1) singled out by the constraints and consists of the triples ’s, where the point transformation is of the form
[TABLE]
with , and , and being arbitrary smooth functions of such that , as well as if is odd and , with if is even.
It is obvious that the subclass is not normalized in the usual sense. Its usual equivalence group is constituted by the point transformations of the form (7) in the joint space of the variables and the arbitrary elements , where parameters satisfy more constraints, , and thus and .
All the components of (7) locally depend on , and, moreover, the expressions for and involve derivatives of with respect to and . This is why, to interpret (7) as generalized equivalence transformations, we need to formally extend the arbitrary-element tuple with the derivatives of as new arbitrary elements, and , , and prolong equivalence transformations to them,
[TABLE]
The derivatives of in the expressions for and should be expanded and then derivatives of should be replaced by the corresponding ’s.
We denote by the class of equations of the form (1) with and the extended arbitrary-element tuple , where the relations defining , …, are assumed as additional auxiliary equations for arbitrary elements.
Theorem 10**.**
The class is normalized in the generalized sense. Its generalized equivalence group coincides with its effective generalized equivalence group and consists of the point transformations in the joint space of the variables and the arbitrary elements with components of the form (7), (8) and the same constraints for parameters as in Proposition 9, where partial derivatives of are replaced by the corresponding restricted total derivatives with and .
Proof.
The point transformations of the above form constitute a group , which generates the entire equivalence groupoid of the class and is minimal among point-transformation groups in the joint space of that have this generation property. Therefore, is an effective generalized equivalence group of the class . We are going to prove that the group coincides with . Indeed, substituting every particular value of to any element of gives an admissible transformation of the class . This implies that elements of are of the form (7), (8), where the parameter functions , and may depend on arbitrary elements, and the partial derivatives of these functions are replaced by the corresponding total derivatives prolonged to the arbitrary elements of the class . At the same time, these parameters satisfy the condition with the prolonged total derivative operator . This condition implies via splitting with respect to unconstrained derivatives of arbitrary elements that the parameters , and are functions of only. Hence . ∎
Therefore, the gauge is better than the gauge .
Remark 11**.**
To the best of our knowledge, Theorem 10 provides the first example for a generalized equivalence group containing transformations whose components for equation variables depend on a nonconstant arbitrary element. This is also an example of a generalized equivalence group being effective itself, and thus the corresponding class of differential equations admits a unique effective generalized equivalence group.
A more complicated example of a generalized equivalence group is given by the subclass of the class (1) singled out by the mere constraint . The -component of equivalence transformations of the class (1) takes the form
[TABLE]
where each , , is a combination of , with coefficients expressed via and derivatives of with respect to . Substituting the expression for implied by the gauge ,
[TABLE]
into the general form of admissible transformations of the class (1) and neglecting the relation between and , we get the elements of the equivalence groupoid of the subclass . Therefore, this subclass is not normalized in the usual sense, and its usual equivalence group is isomorphic to the subgroup of the group singled out by constraining group parameters with . Similarly to Theorem 10, we can consider the counterpart of the subclass , where the tuple of arbitrary elements is formally extended with the derivatives , , and , , . Then the expressions for transformational parts of admissible transformations of and their relations between initial and transformed arbitrary elements including the prolongation to the above derivatives give the components of the transformations constituting a group , which is obviously an effective generalized equivalence group of the class . Thus, the class is also normalized in the generalized sense.
Note that the entire generalized equivalence group of the class coincides with its effective generalized equivalence group . Indeed, in view of the description of the equivalence groupoid of the subclass , elements of are of the form similar to that of elements of , where the group parameters , and may also depend on arbitrary elements of the subclass , and their partial derivatives in and are replaced by the corresponding total derivatives prolonged to the arbitrary elements of the class . At the same time, the condition with the prolonged total derivative operator implies that the parameter functions and still depend at most on . The corresponding expression for involves the derivative , and hence the transformation component for necessarily contains the derivative . Splitting this component with respect to the th order -derivatives of all arbitrary elements, which are not constrained, we derive that in fact the parameter function also does not depend on arbitrary elements.
5 Preliminary analysis of Lie symmetries
We compute the maximal Lie invariance group of an equation from the class (5) using the infinitesimal method. For this, we define the generators of one-parameter point symmetry groups of through with the components , and depending on . The infinitesimal invariance criterion reads
[TABLE]
The th prolongation of the vector field is given by Recall that is a multi-index, , , and . The coefficients in the prolonged vector fields are obtainable from the general prolongation formula [29, Theorem 2.36],
[TABLE]
where , and are the operators of total differentiation with respect to and , respectively, and and .
The infinitesimal invariance criterion yields
[TABLE]
We have shown in Section 3 that the class (5) is normalized in the usual sense. Therefore, we know that the restrictions derived in Corollary 6 for the -components of vector fields from the equivalence algebra also hold for the components of infinitesimal symmetry generators. It is thus true that , , .
Substituting this restricted form of the coefficients of into the infinitesimal invariance criterion (9), we obtain
[TABLE]
This equation can be split with respect to and its spatial derivatives, resulting in the system
[TABLE]
Since all of the determining equations (10) essentially depend on the arbitrary elements , they constitute the system of classifying equations for Lie symmetries of equations from the class (5). Thus, solving the group classification problem for the class (5) reduces to solving the classifying equations (10) up to the equivalence induced by . Due to the structure of the determining equations (10) we have proved the following proposition.
Proposition 12**.**
The maximal Lie invariance algebra of the equation from the class (5) is spanned by the vector fields of the form , where the parameter functions , and run through the solution set of the determining equations (10), and
[TABLE]
Proposition 13**.**
The kernel Lie invariance algebra of equations from the class (5) is trivial, that is, .
Proof.
To derive the kernel of maximal Lie invariance algebras one assumes the arbitrary elements to vary. Then it is possible to split the determining equations (10) also with respect to the arbitrary elements and their derivatives. This immediately yields , and, since and are functions of only, it follows that . ∎
At this stage, it is appropriate to introduce the linear span
[TABLE]
where the parameter functions , and run through the set of smooth functions of . The nonzero commutation relations between vector fields spanning \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} are given by
[TABLE]
In view of these commutation relations, it is obvious that \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} is a Lie algebra. Moreover, it is true that \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}=\cup_{\kappa}\,\mathfrak{g}_{\kappa} since any of the vector fields , and lies in for some particular value of the arbitrary element .
Let us here and in the following denote by the projection map from the joint space of onto the space of the variables alone, i.e., . This map properly pushforwards vector fields from and transformations from , and \pi_{*}\mathfrak{g}^{\sim}=\mathfrak{g}_{\mbox{\tiny\langle,\rangle}}. This displays that in fact the class (5) is strongly normalized in the usual sense [36]. Note that the normalization of the class (5) in the usual sense only implies that \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}\subseteq\pi_{*}\mathfrak{g}^{\sim}. The spanning vector fields of , , and , are pushforwarded by to the spanning vector fields , and of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}, respectively. By means of the pushforward by , the adjoint action of on induces the action of on \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} and, therefore, on the set of subalgebras of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}. Recall that a subalgebra of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} is called appropriate if there exists a such that . For any value of the arbitrary element and any transformation , the pushforward by maps the maximal Lie invariance algebra of the equation onto the maximal Lie invariance algebra of the equation , and both the algebras and are included in \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}. In other words, the action of on \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} preserves the set of appropriate subalgebras of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}, and hence the group generates a well-defined equivalence relation on these subalgebras. As a result, we have proved the following proposition, which is the basis for the group classification of class (5).
Proposition 14**.**
The complete group classification of the class (5) of gauged -dimensional general Burgers–KdV equations of order is obtained by classifying all appropriate subalgebras of the Lie algebra \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} under the equivalence relation generated by the action of .
To efficiently carry out the group classification using the algebraic method, it is necessary to compute the adjoint actions of the transformations from on the vector fields from \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}.
The adjoint actions of the transformations, , from on the vector fields, , from \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} are directly computable from the definition of the pushforward of by ,
[TABLE]
see, e.g., [2, 3, 6, 19]. Here, the coefficients of are given in terms of the transformed variables obtained by substituting using the inverse transformation of .
In practice, this is done by considering the families of elementary transformations , and from , which follow from (6) by restricting all except one of the parameter functions , and to trivial values (which are for , one for and zero for ). The nontrivial pushforwards of the spanning vector fields of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} by these elementary transformations from are given by
[TABLE]
Here the tildes over the right-hand side operators indicate that the given vector fields are expressed using the transformed variables, which also includes substituting for , where is the inverse function of .
6 Properties of appropriate subalgebras
An important step for the group classification of class (5) is to study properties of appropriate subalgebras of the algebra \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} to be classified. In particular, we determine the maximum dimension of admitted Lie invariance algebras of equations from this class. This is formulated in the following lemma.
Lemma 15**.**
For any tuple of arbitrary elements , .
Proof.
This statement follows directly from analyzing the solution space of the linear system (10) with respect to , and for a fixed tuple . Denote by and open intervals on the - and -axes, such that the equation is defined on the domain . Since by definition, we can resolve the equation (10a) with for and fix , yielding
[TABLE]
If the arbitrary element is fixed, the system (12) can be considered as a canonical system of linear ordinary differential equations in for , and . Its solution space is thus five-dimensional. Further conditions derived from the classifying equations (10) can only reduce this solution space and hence it follows that . ∎
We introduce three integers related to the dimensions of certain subspaces of the maximal Lie invariance algebra of the equation ,
[TABLE]
Although these integers depend on , in view of (11) it is obvious that the dimensions of the subalgebras and as well as of the entire algebra are -invariant, and thus the integers ’s are also -invariant.
Lemma 16**.**
The integers , and are -invariant values, i.e., they are the same for all -equivalent equations from the class (5).
Proof.
Let transform to . The transformation pushforwards onto . At the same time, it preserves the spans and . This is why , and . ∎
We now proceed to find the upper bounds for values of these integers, which is proved in a way similar to Lemma 15.
Lemma 17**.**
For any , we have .
Proof.
For any vector field from the algebra , the parameter functions and satisfy the system of classifying equations (10) for the chosen tuple of arbitrary elements and .
In particular, if , we can solve the classifying equation (10a) with with respect to to obtain . After fixing a value , this equation implies that there exists a function such that . Then the classifying equation (10b) implies that . Again fixing , we hence derive an equation with some function . Since the parameter functions and of any vector field from satisfy the same system and , we have and in this case.
If , then the classifying equation (10a) with directly gives , i.e., . Suppose that , i.e., there exists a nonzero such that . It then follows from the system (10) that , and . Differentiating the last equation with respect to results in . Thus, we have for any from the two-dimensional solution space of the equation , which means . ∎
The projection from the space of on the space of alone properly pushforwards elements of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} according to , and hence \varpi_{*}\mathfrak{g}_{\mbox{\tiny\langle,\rangle}}=\{\tau\partial_{t}\}, where runs through the set of smooth functions of . The pushforward of by is also well defined.
Lemma 18**.**
The projection is a Lie algebra for any tuple of arbitrary elements , and . It is true that .
Proof.
We show that is indeed a Lie algebra. Given , , there exist such that . For any constants and , we have that and thus . This means that is indeed a linear space. This space is closed under the Lie bracket of vector fields and, therefore, is a Lie algebra, because .
Since the pushforward of by the projection coincides with the (pseudo)group of local diffeomorphisms in the space of , Lie’s theorem can be invoked. It states that the maximum dimension of finite-dimensional Lie algebras of vector fields on the complex (resp. real) line is three. Up to local diffeomorphisms of the line, these algebras are given by , , and . ∎
It follows that .
7 Group classification
The main result of the paper is given by the following assertion.
Theorem 19**.**
A complete list of -inequivalent (and, therefore, -inequivalent) Lie symmetry extensions in the class (5) is exhausted by the cases given in Table 1.
Proof.
Lemmas 17–18 imply that any appropriate subalgebra of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} has a basis consisting of
vector fields , , with linearly independent ’s, 2. 2.
vector fields , , with nonzero ’s, 3. 3.
vector fields , , with linearly independent ’s,
where and . The proof proceeds by separately investigating the cases associated with the possible range of the tuple of invariant integers . For each possible value of , we start with the above form of basis vector fields ’s of and simplify them as much as possible using the adjoint actions of equivalence transformations presented in (11) and linear recombination of ’s. At the same time, we take into account the fact that is a Lie algebra, i.e., it is closed with respect to the Lie bracket of vector fields, , . This leads to constraints for the components of ’s only if or and . Substituting the components of each simplified into the system of classifying equations (10) yields a system of equations in for which . In total, we have such systems. We unite them and simultaneously solve for the arbitrary elements . The compatibility of the joint system with respect to may imply additional constraints for the components of ’s. The expression obtained for can be simplified by equivalence transformations whose projected adjoint actions preserve . Except Cases 1 and 1, the condition for with is obtained by the negation of the corresponding condition represented in Remark 22.
We have to consider the following cases:
Here . For , in view of Lemma 18 we can use the simplified form of , , derived in the subcase with the preceding value of .
. We obtain the general Case 1, where the algebra coincides with the kernel algebra and thus there are no constraints for .
. A basis of consists of a single vector field with . Successively using the adjoint actions , and given in (11) for appropriate functions , and , this vector field can be mapped to . The system of classifying equations (10) with the components of evidently is , which results in Case 1 of Table 1.
. Modulo -equivalence, basis elements of take the form and . The condition requires that and thus . Substituting the components of and then into the classifying equations (10) and rearranging leads to the system
[TABLE]
If , in view of the condition the equation implies that and we can set ; otherwise, we can set using , where , which preserves . Integrating the system (13) for each of the subcases, we respectively obtain Cases 1 and 1, where and thus . In Case 1, one of the constants ’s with , or , if it is nonzero, can be set to by shifts of .
Modulo -equivalence and linearly combining basis elements, we can assume that , and . The completeness of the algebra with respect to the Lie bracket of vector fields implies that , and . As in the previous case, the first commutation relation holds only if . The second commutation relation expands to
[TABLE]
and thus , . Integrating these two equations yields and , where and are constants. Next, commuting and yields
[TABLE]
We thus have if and only if and . Considering the classifying equations (10) for , and , we obtain the system (13) supplemented by the equations , and . If , then , , and the equations imply that for any , which contradicts the condition . Therefore, , which requires that , as well as . We can then set using . The remaining determining equations are and , which are readily integrated to Case 1 with , and hence .
Hence , where .
. We can use the adjoint action to set . Moreover, multiplying by a nonzero constant and changing if is not a constant, we can gauge to , where . The classifying conditions (10) with components of then imply the system consisting of the equations , and . The general solution of this system is
[TABLE]
where , and are arbitrary smooth functions of with . The subgroups of the equivalence group whose projections to the space preserve the appropriate subalgebras and are singled out from by the constraints and , respectively. Elements from these subgroups allow us to set if or if , which corresponds to Case 1 or 1. In the latter case we have chosen the gauge instead of in order to make the corresponding values of the tuple of arbitrary elements simpler.
. First we reduce, modulo -equivalence, the basis element with to the form . The Lie bracket is in provided that the tuples and are linearly dependent, i.e., , for some constants , and , where . Multiplying by and, if , using , we can set and . Since and , we can finally reduce to the form . Therefore, the corresponding complete system for the arbitrary elements includes the system from the previous case as a subsystem supplemented by the equations . The general solution of the complete system is of the same form as in the case , where the functions , and should be replaced by the arbitrary constants , and with . Similarly to the case , the appropriate subalgebras and are preserved by projections of equivalence transformations with and , which makes possible the gauges or if or and obviously results in Cases 1 and 1, respectively.
. This case cannot be realized. Indeed, otherwise we would additionally have, modulo -equivalence, the basis element and would still be able to repeat the consideration of the case , deriving the expression with . At the same time, the classifying equation (10a) for with the components of and with the above reads , which would imply , giving a contradiction.
Here we have , . The classifying equations (10) in this case imply that , and . Differentiating this last equation with respect to leads to , i.e., , and therefore .
. We apply to and set , which implies that . Then using the adjoint action we can set . The equations , , then imply that . The components of the transformation for the arbitrary elements are , and . Thus, choosing allows us to gauge , leading to Case 1.
. We start the simplification of basis elements of from with , setting, modulo -equivalence, . It then follows from the classifying equations (10) with the components of that . We choose
[TABLE]
Since , , , and for the chosen value of we have , the adjoint action preserves the algebra . At the same time, the component of for is and therefore , which yields Case 1, where .
. We have one more basis element whose simplified form is . Commuting and yields , which is in the algebra provided that . The commutator of with , , gives which is in the algebra only if and up to linearly combining and . It then follows that . We can let , and the classifying equation (10c) then implies that . Upon linearly combining with and we can thus set . Moreover, the classifying equation (10a) for with the components of implies that and thus, in view of the same equation for other ’s, if . We can gauge . Now suppose that we also have a . Then, the classifying equations (10) for and require that and . This system is consistent only for , where . This is why Case 1 splits depending on the value of . The equation (10c) with implies that and thus we can set upon linearly combining with and . ∎
Corollary 20**.**
An th order evolution equation of the form (1) is reduced to the simplest form by a point transformation if and only if the dimension of its maximal Lie invariance algebra is greater than three.
Remark 21**.**
Case 1 can be merged with Case 1 as the particular subcase with if we choose another second basis element, .
Remark 22**.**
Each of the subalgebras of whose bases are presented in the third column of Table 1 is really the maximal Lie invariance algebra for the general case of values of the arbitrary elements given in the same row. For most of the classification cases, it is not difficult to explicitly indicate the necessary and sufficient conditions for under which there is an additional Lie symmetry extension. They can be found by substituting the corresponding expressions for ’s components into the system of classifying conditions (10). Thus, these conditions are if or with if with in Case 1; and in Case 1; in Cases 1 and 1; for and in Case 1. There are no additional extensions in Cases 1, 1, 1 and 1. An additional extension exists for Case 1 if and only if the parameter functions ’s satisfy the system for some constants , , and . Another form of this condition, which is convenient for checking, is that
[TABLE]
for some functions and constrained by , . See also the claim on the most complicated Case 1 in Remark 35.
Remark 23**.**
The proof of Theorem 19 shows that the system of restrictions for appropriate subalgebras of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} presented in Lemmas 15, 17 and 18 is not exhaustive. It can be completed by the conditions restricting the value set of depending on values of and ,
[TABLE]
For each value of the tuple satisfying the extended set of restrictions, there exists an appropriate subalgebra of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}} admitting this value of .
8 Alternative classification cases
There are various possibilities for choosing representatives in equivalence classes of pairs , where is an appropriate subalgebra of . We have tried to simplify the representation of a pair by paying more attention to the pair’s second entry. In most cases the optimal choice is obvious and coincides with the selection carried out for Table 1. At the same time, there are other options for the proof and representation of results in the cases and .
We follow the proof of Theorem 19 but reduce the basis vector fields to another form. If , we set, modulo -equivalence, instead of , where still . In the case , for we choose the form in order to be able to set again. For , this gives the following alternative cases, which are related to the corresponding cases of Table 1 by the equivalence transformation with , and :
[TABLE]
Instead of setting in the proof for this case, we can simplify the basis elements and to and as in the case . Then readily , and, modulo -equivalence, . Consider the subclass of equations from the class (5) with values of satisfying the above constraints,
[TABLE]
The subclass turns out to be normalized with respect to its usual equivalence group , which is finite-dimensional and consists of the transformations in the space of whose components are of the form (6a)–(6b) with
[TABLE]
where , …, are arbitrary constants with that are defined up to a nonzero multiplier. Hence the equivalence algebra of is spanned by , , , , and . The kernel Lie invariance algebra of is . The restriction of -equivalence to the subclass coincides with -equivalence. This is why, up to -equivalence, we can assume that in this case the appropriate subalgebra is spanned by , and , and there are three cases for and :
(a) , , ;
(b) , , ;
(c) , , .
Therefore, instead of the single Case 1 we have the following three cases:
[TABLE]
Modulo -equivalence, , and in Cases , and , respectively.
The advantage of this form for Case 1 is that the vector fields and then have the evident interpretation as generators of translations with respect to the space variable and Galilean boosts, respectively. Meanwhile the forms of and, especially, become more complicated. This is why we chose the previous form of Case 1 for Table 1. Subcases of this case are related to subcases of the alternative Case by equivalence transformations of the form (6), where and
[TABLE]
9 Equations with time-dependent coefficients
To study Lie symmetries of equations from the class (1) with coefficients depending at most on , it is again convenient to start with a wider class, which is the subclass of the class (1) singled out by the constraint (resp. ) implying for admissible transformations.
Proposition 24**.**
The class is normalized in the usual sense. Its usual equivalence group is constituted by the transformations of the form
[TABLE]
where , and , , , and are arbitrary smooth functions of their arguments with .
Consider the subclass obtained by attaching the constraints , , , , and to the auxiliary system for arbitrary elements. It is also normalized in the usual sense and its usual equivalence group is the subgroup of the usual equivalence group of the class that is associated with the constraint , i.e., . Note that we can reparameterize the class by representing , and assuming the coefficients , , and as arbitrary elements instead of and . The transformation component for simplifies to
[TABLE]
The next intermediate subclass is singled out by strengthening the constraint for to . In fact, this can be realized by gauging in the class up to -equivalence. The transformational properties of the subclass are similar to those of the subclasses studied in Section 4, which is related to bringing the constraint for into the foreground among the constraints or gauges for other arbitrary elements; see further discussions below. Since the arbitrary element is still not gauged to one, it parameterizes the -component of admissible transformations in , and this fact can again be interpreted in terms of generalized equivalence group.
[TABLE]
Theorem 25**.**
The equivalence groupoid of the subclass of the class (1) singled out by the constraints , , and consists of the triples ’s, where the point transformation is of the form
[TABLE]
the arbitrary-element tuples and are related according to
[TABLE]
with , and , , , and are arbitrary smooth functions of with .
The usual equivalence group of the subclass is constituted by the transformations (14a)–(14d) additionally satisfying the constraint . Hence it is clear that the subclass is not normalized in the usual sense.
The equation (14c) hints that the proper treatment of the related generalized equivalence group within the framework of point transformations needs considering the derivative as an additional arbitrary element and prolonging the relation (14d) to as a derivative of ,
[TABLE]
We denote by the class in which the tuple of arbitrary elements is formally extended to with .
Corollary 26**.**
The class is normalized in the generalized sense. The group constituted by the transformations of the form (14) is an effective generalized equivalence group of this class.
Proof.
The set of the transformations of the form (14), which is temporarily denoted by , is closed with respect to the transformation composition and contains the identity transformation. Each transformation from is invertible by definition. So, is a group. The components of transformations from are of the same form as the components of admissible transformations and the formulas relating the initial and target arbitrary elements. This is why the group generates the equivalence groupoid and, moreover, it is minimal among subgroups with such property. Therefore, is an effective generalized equivalence group of the class . ∎
The entire generalized equivalence group of the class is much wider than its effective part .
Corollary 27**.**
The generalized equivalence group of the class consists of the transformations of the modified form (14), where , , , and are arbitrary smooth functions of their arguments with , and the partial derivatives of , and in should be replaced by the corresponding restricted total derivatives in with .
Proof.
Theorem 25 implies that elements of are of the modified form (14), where the group parameters , , , and may depend on and the arbitrary elements . Hence partial derivatives of these parameter functions should be replaced by the corresponding total derivatives in with
[TABLE]
After substituting for the derivative , the transformation components can be split with respect to the other derivatives of arbitrary elements in . The splitting implies that in fact the group parameters do not depend on ’s, and , and, moreover, the parameters and do not depend on . The nondegeneracy condition for elements of is modified in comparison with that for elements of the effective part in view of the parameter function becoming dependent on . This condition takes the form and reduces to the condition given in the statement of the theorem. ∎
Remark 28**.**
Given a class of differential equations with nontrivial effective generalized equivalence group, this group is in general not defined in a unique way. Indeed, consider the class . The effective generalized equivalence group defined in Corollary 26 is not a normal subgroup of the entire generalized equivalence group of the class . Each subgroup of that is conjugate to is an effective generalized equivalence group of the class . In other words, the class possesses a wide family of conjugate effective generalized equivalence groups. The similar fact is even more obvious for the class studied below.
To have the required subclass of equations from the class (1) whose coefficients depend at most on , we now only need to impose a more restrictive constraint on , replacing the additional auxiliary equation by , which can be implemented by gauging within the class using its equivalence transformations. Unfortunately, this deteriorates the normalization property since then the function parameterizing elements of the equivalence groupoid of the class depends on the initial arbitrary elements and in a nonlocal way via the equation
[TABLE]
At the same time, the usual equivalence group of the subclass coincides with the group . The computation of the generalized equivalence group of the subclass gives the same group, which is a trivial situation from the point of view of generalized equivalence. As a result, the class is definitely not normalized in both the usual and the generalized senses. This is why we construct the extended generalized equivalence group of the subclass in a rigorous way. In fact, this is the first construction of such kind in the literature.
We extend the arbitrary-element tuple to with two more arbitrary elements, and , which are functions of only and satisfy the auxiliary equations
[TABLE]
Thus, we also implicitly impose the auxiliary equations , , . Each value of satisfying all auxiliary equations of the class as well as the above equations for and is associated with an equation of the form (1) with the corresponding value of . We formally denote this equation by and the class of such equations by . It is obvious that the equations and coincide if . This defines a gauge equivalence relation on the value set of arbitrary-element tuple . We show below that this gauge equivalence gives rise to a nontrivial gauge equivalence group of the class . (See Sections 2.1 and 2.5 of [36] for notions related to gauge equivalence, which is called trivial equivalence in [22].) Since the set of point transformations from to coincides with that from to , the equivalence groupoid of is isomorphic to the equivalence groupoid of factorized with respect to the gauge equivalence. In the class , the constraint (15) can be solved with respect to in terms of ,
[TABLE]
where and are arbitrary constants with . Using this solution and the auxiliary equations (16), we prolong the relation (14b)–(14d) between initial and transformed arbitrary elements to and . Thus, the equality chain
[TABLE]
implies for some constant . Considering the equality chain
[TABLE]
where , we derive for some constants and with that
[TABLE]
We use parentheses instead of vertical bars in the logarithm since . This completes the description of the equivalence groupoid . Note that here
[TABLE]
Theorem 29**.**
Let be the subclass of equations from the class (1) with coefficients depending at most on , which is singled out from the class (1) by the constraints , . The class of the same equations, where the arbitrary-element tuple is formally extended with the virtual arbitrary elements and defined by (16), is normalized in the generalized sense. Its generalized equivalence group consists of the transformations of the form
[TABLE]
where ; , , and are arbitrary smooth functions of , and with ; , , and are arbitrary constants with and, moreover, ; is the restricted total derivative operator with respect to .
Proof.
In view of the above description of the equivalence groupoid of the class , elements of have the general form
[TABLE]
The computation of by the direct method is quite similar to the computation of and, after splitting with respect to and parametric derivatives of , gives similar expressions for transformation components for the variables and similar constraints for parameter functions. The relations between the initial and target arbitrary elements in the equivalence groupoid just convert to the transformation components for arbitrary elements in the equivalence group. But there are several differences, which we are going to discuss.
In particular, the total derivative operators should be prolonged to the arbitrary elements. Since the arbitrary elements of the class depend at most on , the prolongation is essential only for ,
[TABLE]
The expression for is formally preserved, . As a result, all partial derivatives with respect to in the expressions derived after splitting with respect to and parametric derivatives of are converted to the total derivatives with respect to .
The second difference is the possibility of splitting with respect to arbitrary elements and their derivatives. After substituting for the constrained derivatives and in view of (16) into the constraint for ,
[TABLE]
we can split the resulting equation with respect to , …, , , , and . This leads to the system , , , , and , whose general solution is of the form (17). The expressions for the transformed arbitrary elements , …, , and can also be split with respect to unconstrained derivatives of arbitrary elements in , implying that the derivatives of , , and with respect to , …, , and are zero. Hence the operator can be replaced by the restricted total derivative operator . In particular, the parameter function is defined by .
The additional auxiliary equations (16) are also treated in a different way. We substitute the expressions for and given by these equations into their expanded version for transformed arbitrary elements. Splitting the resulting equations with respect to the other derivatives of arbitrary elements leads to the system of determining equations for the -components of equivalence transformations
[TABLE]
whose general solution is of the form presented in the statement of the theorem. ∎
Remark 30**.**
Each element of the generalized equivalence group generates a family of admissible transformations of the class with sources at those values of where the evaluation of does not vanish,
[TABLE]
Here is the value set of the arbitrary-element tuple of the class .
The gauge equivalence group of the class is the subgroup of that is singled out by the constraints , , , , , . In other words, all the components of gauge equivalence transformations are identities, except the components for and , for which we get , with . The usual equivalence group of the class is singled out from by the constraints
[TABLE]
and its quotient group with respect the gauge equivalence group of the class is isomorphic to the usual equivalence group of the class .
It is obvious that the generalized equivalence group of the class generates the whole equivalence groupoid of this class. At the same time, functions parameterizing the group depend on two more arguments, and , than functions parameterizing the groupoid do. If we omit the arguments and in the parameter functions, the corresponding set of transformations still generates the equivalence groupoid but it is not a group with respect to the transformation composition. This shows that the class may possess an effective generalized equivalence group being a proper subgroup of , and its construction needs a more delicate consideration than, e.g., for the class .
Corollary 31**.**
The class is normalized in the extended generalized sense. Its extended generalized equivalence group can be identified with the effective generalized equivalence group of the class that consists of the transformations of the form
[TABLE]
where ; and , , and are arbitrary smooth functions of with ; , , and are arbitrary constants with and, moreover, .
Proof.
We temporarily denote by the set of the transformations of the above form. This set is a subset of the group . It is singled out from by setting the following values for group parameters:
[TABLE]
The set is closed with respect to the transformation composition, i.e., is a subgroup of the group .
The subgroup generates the entire equivalence groupoid of the class and thus the entire equivalence groupoid of the class . Indeed, let us fix any equation from the class . The set of all admissible transformations with source at is parameterized by the arbitrary smooth functions , , and of and the arbitrary constants , , and with , and , where is defined by (17) for the fixed value of the arbitrary element , . Each admissible transformation from is generated by the equivalence transformation from with the same values of , , , and , and the values of , and defined by
[TABLE]
This establishes a one-to-one correspondence between and , and thus the subgroup is minimal among the subgroups of that generate the groupoid .
Therefore, is the effective generalized equivalence group of the class . ∎
Nevertheless, we can further gauge arbitrary elements of the class in such a way that the corresponding subclasses of have better normalization properties. Up to -equivalence we can set, as above, and . Moreover, since the class is parameterized by functions depending at most on , -equivalence allows us to further gauge arbitrary elements by setting and , which results in the class normalized in the usual sense and whose usual equivalence group is finite-dimensional, see Section 8. At the same time, gauging arbitrary elements by usual equivalence transformations in a class that is not normalized in the usual sense does not allow one to easily control the changing of the corresponding equivalence groupoid. This is why we revise the order of gauging in order to construct a subclass normalized in the usual sense for each step of gauging.
Remark 32**.**
In general, if a multiple gauge is realized step-by-step, then the order of steps is also important and intermediate subclasses may have inferior normalization properties than the original class and the final gauged subclass.
We use the following order of gauging for singling out the subclass from the class : , , , and , which successively constrains the group(oid) parameters by
[TABLE]
Note that, e.g., the ordering of the last two gauges is not essential but if we try to carry out at least one of them before the gauge , then we obtain a subclass normalized in the generalized extended sense; cf. the derivation of Corollary 31. Thus, the parameter function of the equivalence groupoid for the subclass gauged with , and is related to the initial value of the arbitrary element via the equation (15) with .
Finally, the solution of the group classification problem for the class up to the generalized extended -equivalence reduces to that for the class up to the usual -equivalence. Combining results of Sections 7 and 8, we readily obtain the following assertion.
Theorem 33**.**
A complete list of -inequivalent Lie symmetry extensions in the class (resp. -inequivalent Lie symmetry extensions in the class ) is exhausted by Cases 1, , , and 1 given in Table 1 and Section 8.
Whereas each equation from the class admits the two-dimensional algebra with and constituting a fundamental set of solutions of the linear ordinary differential equation , the kernel Lie invariance algebra of this class is zero since the functions and depend on the arbitrary elements and . The situation with the class is different: its kernel Lie invariance algebra is .
10 Equations with space-dependent coefficients
The subclass of equations with space-dependent coefficients is singled out from the class (1), which is normalized in the usual sense, by the constraints , , and . This is why the usual equivalence group of the subclass is a subgroup of the equivalence group of the class (1) that consists of transformations preserving the above constraints. In view of Proposition 2, the -components of transformations from are of the form
[TABLE]
where , , and are smooth functions of their arguments such that . (It is convenient here to conversely express source arbitrary elements via target ones.) Differentiating the - and -components with respect to and taking into account the additional constraints for arbitrary elements of the subclass , we derive classifying equations for admissible transformations in ,
[TABLE]
which should be split with respect to derivatives of arbitrary elements when constructing usual equivalence transformations of the subclass . Hence for these transformations. Then treating the -component in the same way gives one more condition for group parameters, . In fact, the obtained conditions for group parameters constitute the complete system of such conditions that singles out the group as a subgroup of the group .
Proposition 34**.**
The usual equivalence group of the class of general Burgers–Korteweg–de Vries equations with space-dependent coefficients consists of the transformations in the joint space of whose -components are of the form
[TABLE]
where , and are arbitrary constants and and are arbitrary smooth functions of such that .
The existence of classifying conditions for admissible transformations means that the class is definitely not normalized in any sense. This is why a proper starting point for computing the equivalence group of a subclass of is not the equivalence group of but the equivalence group of the superclass (1) or of a normalized subclass of (1) that contains .
Two parameters of the group of the class are of the same arbitrariness as the arbitrary elements of this class. Therefore, we can set two gauges for the arbitrary elements of using parameterized families of transformations from . Following the consideration of the superclass (1) in Section 3, we successively gauge to 1 and to 0.
In view of the expression for the -component of transformations from , the gauge can really be implemented. We denote the corresponding subclass of by . The usual equivalence group of the subclass turns out to be the subgroup of singled out by the constraint , i.e., and . For transformations from , the -component can be neglected and the expression for the -component simplifies to . It becomes clear that the further gauge is also realizable, which gives a subclass of , and we denote it by . The usual equivalence group of this subclass turns out to be the subgroup of the group singled out by the constraint . In other words, the group is just four-dimensional and consists of the transformations of the simple form
[TABLE]
where , and , …, are arbitrary constants with . The -component can be neglected.
Since the subclasses and are constructed via gauging the arbitrary elements of the class by usual equivalence transformations, and , then any classification problem for the class up to -equivalence (like the group classification problem or the description of the corresponding equivalence groupoid) reduces to the respective classification problem for the class up to -equivalence, cf. Proposition 7.
We begin with the description of the equivalence groupoid of the class . Since the class is not normalized and neither are its superclasses and , this problem should be considered as the classification of admissible transformations up to -equivalence. See Section 2.6 and 3.4 in [36]. Since the class is a subclass of the class (5), the equivalence groupoid is a subgroupoid of the equivalence groupoid of the class (5). Therefore, all the restrictions from Theorem 5 on the form of admissible transformations are relevant here. We solve the relations (6b)–(6c) with respect to the source arbitrary elements, and differentiate the result with respect to . This gives the classifying conditions for admissible transformations in terms of target arbitrary elements,
[TABLE]
where the initial space variable should be substituted, after expanding all derivatives, by its expression via , . Since admissible transformations with are generated by the usual equivalence group , the problem on classifying of admissible transformations can be stated in the following way: To find -inequivalent values of the arbitrary-element tuple for which the system (19) with respect to has solutions with .
For each fixed , the system (19) implies a system of ordinary differential equations with respect to the arbitrary-element tuple of the form
[TABLE]
where , …, are constants. For each value of the arbitrary-element tuple , we denote by the span of the set through which runs when the tuple takes values possible for elements of with the target and the variable is varied. It is obvious that . The consistency of system (19) with the condition implies that . Therefore, the equality means that for all admissible transformations in with target at . The further consideration partitions into five principal cases, depending on the value and certain constraints for components of nonzero , I) , II) , , III) , , , IV) , , with and V) . For each of these cases, we integrate the corresponding set of independent copies of the system (20) and thus derive the respective form of parameterized by arbitrary constants. The relations (6b)–(6c) between and imply that the arbitrary-element tuple is of the same form as , but maybe with other values of parameter constants. In other words, subclasses associated with the above classification cases are invariant under the action of the equivalence groupoid , which additionally justifies the chosen partition into cases. We substitute the derived expressions for and into the relations (6b)–(6c) and, assuming , split the resulting equations with respect to . When integrating the obtained system of determining equations for the parameter functions , and of point transformations mapping the equation to the equation , we check for the existence of solutions with . This implies constraints for constants parameterizing the expressions for , which is in fact identical to the similar constraints for . Finally, we gauge constant parameters of arbitrary elements with transformations from , where this is possible, and show that the lists of -inequivalent cases of admissible-transformation extensions and of Lie symmetry extensions in the class coincide.
Note that a complete list of -inequivalent Lie symmetry extensions in the class (resp. -inequivalent Lie symmetry extensions in the class ) is exhausted by Cases 1, 1, 1, 1, 1, 1, 1 and 1 of Table 1. This follows from the facts that the integer , which is defined in Section 6, is -invariant when considered for equations from the class and is greater than or equal to 1 for these equations. Case 1 of Table 1 is the general case for the class . In other words, the kernel Lie invariance algebra of the class is .
Now we list the subclasses of with admissible-transformation extensions. For each of the subclasses, we present the corresponding form of arbitrary elements, the description of admissible transformations and the equivalent case of Table 1. Below ; ’s and ’s are constants parameterizing the arbitrary-element tuple within the subclass under consideration, and ’s are constants parameterizing admissible transformations.
I. , , modulo shifts of , where ; if ; if ; , , and . Then (due to the above shifts of ), , , , , and . .
a) If , then the expression for is given by the general solution of the equation with the constants and defined by
[TABLE]
(See [40] for a proper representation of this solution.) Since , this case reduces to Case 1 of Table 1.
b) If , then , and the expression for is given by the general solution of the equation
[TABLE]
Since , this case reduces to Case 1 of Table 1.
II. , , modulo shifts of with , and . Then (due to the above shifts of ), with , , , , and runs through the set of (nonzero) solutions of the equation
[TABLE]
If , then and , giving Case 1 of Table 1. Otherwise, and , i.e., we obtain Case 1 of Table 1.
III. , , , where , and . Then , , , , , , , the expression for is given by the general solution of the equation . We can gauge and , which leads to Case 1 of Table 1. One of the constants ’s with , or , if it is nonzero, can be set to by shifts of .
IV. , , with , where also or for some . Then with , , , , , and runs through the solution set of the equation . Gauging and , we reduce this case to Case 1 of Table 1.
V. , , , , with . We have , , and the parameter function runs through the solution set of either the equation
[TABLE]
if or , respectively. The expression for is found from the equation
[TABLE]
For any allowed value of , we have and thus get Case 1 of Table 1.
There are no additional -gauges of constants parameterizing the arbitrary-element tuple .
By we denote the subclass of equations from the class whose maximal Lie invariance algebras are similar to the algebra presented in Case of Table 1, . In this way, we partition the class into eight subclasses, that are associated with cases of Lie symmetry extensions in this class. The seven subclasses , , can be shown to be normalized in the generalized sense, and the complement to their union, which is the subclasses , is normalized in the usual sense with respect to the usual equivalence group of the entire class . There are no point transformations between equations from the subclasses and if . In other words, the equivalence groupoid of the class is the disjoint union of the equivalence groupoids of the normalized subclasses ’s.
Remark 35**.**
The above consideration implies that the Lie symmetry extension given in Case 1 of Table 1 is maximal if and only if the arbitrary-element tuple is not of one of the forms listed in Cases I–V.
11 Lie reductions and exact solutions
Since the class (1) is mapped onto its subclass (5) by a family of equivalence transformations and both these classes are normalized, for studying Lie reductions of equations from the class (1) we can apply the technique suggested in [32] to the subclass (5). Within the framework of the standard approach to Lie reduction, one considers each case of Lie symmetry extensions separately, lists subalgebras of the corresponding maximal Lie invariance algebra that are inequivalent with respect to internal automorphisms of this algebra, constructs ansatzes with them and then uses these ansatzes to reduce relevant original equations to equations with fewer independent variables. For a normalized class of differential equations, one can classify Lie reductions of equations from the class with respect to its equivalence group of respective kind, taking subalgebras in the projection of its equivalence algebra [32]. Restrictions for appropriate subalgebras are relevant here except those restrictions related to the property to be maximal for some equations from the class.
A complete list of -inequivalent one-dimensional subalgebras of the algebra is exhausted by four subalgebras, , , and , cf. Table 1. The subalgebra does not appear in Table 1 since, in view of Lemma 17, there is no equation from the class (5) for which this algebra is the maximal Lie invariance algebra. This subalgebra is associated with Case 1 of Table 1. We construct an ansatz for with each of the listed one-dimensional subalgebras, select -inequivalent equations from the subclass (5) that are invariant with respect to this subalgebra and reduce them to ordinary differential equations. In the course of selecting invariant equations, we use Table 1 but neglect the conditions of maximality of invariance algebras, which are collected in Remark 22. As a result, we obtain the following -inequivalent reductions to ordinary differential equations:
[TABLE]
Now we classify two-dimensional subalgebras of that are subalgebras of appropriate subalgebras of and carry out corresponding Lie reductions of equations from the class (5) to algebraic equations. To obtain a complete list of such subalgebras up to -equivalence, we realize the following algorithm: take two operators of the most general form, verify that they satisfy necessary conditions and then simplify successively their form until all cases are listed. Computations from the proof of Theorem 19 can be used here. At first, let two vector fields and be of the form , , where the parameter functions and are linearly independent. As has already been shown in the case of the above proof, this case is exhausted by the subalgebras and , where . Suppose that and are linearly dependent but are not simultaneously zero. Then, up to linearly re-combining and , we can assume that , and, moreover, the first basis vector field can be transformed to . If in the new , the consideration is entirely reduced to the case of the proof of Theorem 19, which gives the subalgebras and . Otherwise we have two other inequivalent possibilities for , and . The basis elements cannot be of the form , , with in view of Lemma 17. And finally, the subalgebra spanned by , , with linearly independent parameter functions and does not provide a Lie ansatz for . Therefore, -inequivalent reductions of equations from the class (5) to algebraic equations are exhausted by the following reductions:
[TABLE]
Here is the falling factorial.
It is obvious that the Burgers, Korteweg–de Vries, Kuramoto–Sivashinsky and Kawahara equations admit solutions affine with respect to , which are of the form or with arbitrary constants and . This is also true for generalized Burgers equations of the form , which is related to the fact that each of these equations is conditionally invariant with respect to the vector field [31, 32] or, equivalently, with respect to the generalized vector field . The above Lie reductions with respect to the algebras and also solely give solutions affine in . This is why we can look for such solutions of equations from a wide subclass of the class (1). More specifically, each equation of the form (1) with , i.e., , and , is reduced by the ansatz to a system of two ordinary differential equations for and ,
[TABLE]
This means that the equation possesses the generalized conditional symmetry , and the above ansatz just represents the general solution of the corresponding invariant surface condition ; cf. [48] as well as [16]. If and , then solutions constructed with the ansatz affine in are of the same form as for the Burgers and Korteweg–de Vries equations.
All the solutions constructed here are quite simple. Perhaps the most direct way to construct nontrivial solutions for variable-coefficient equations from the class (1) is to generate them by equivalence transformations from wide families of solutions known for famous (constant-coefficient) equations from the same class, like the Korteweg–de Vries equation. This way is often ignored in the literature, which is comprehensively discussed in [37].
12 Conclusion
The class (1) is much wider than the classes of variable-coefficient Korteweg–de Vries and Burgers equations, which were considered in [10] and [38] and are the subclasses of (1) for and with the constraints and , respectively. At the same time, we have carried out the complete group classification of the class (1) with less efforts than the ones made in [10, 38] for the group classification of the above particular subclasses, and the classification list is much more compact for the class (1), although the equation order is not fixed here. This became possible due to using several ingredients.
The first ingredient was the proper choice of the class to be studied. In general, the choice should be realized in such a way for the selected class to be convenient for group classification, with the main criterion for this being the property of normalization. In contrast to the classes considered in [10] and [38], the class (1) is normalized in the usual sense.
Since there are two arbitrary smooth functions of among the parameters of the usual equivalence group of the class (1) and arbitrary elements of this class also depend on only, two of the arbitrary elements can be gauged by equivalence transformations. It seems at first sight that there are several equally appropriate gauges for arbitrary elements of the class (1) but this is not the case. It has been shown in Section 4 that although the gauge is quite natural and gives a subclass normalized in the usual sense, the structure of the corresponding usual equivalence group is more complicated than for the gauge , and the further possible gauge leads to a subclass that is normalized solely in the generalized sense. The equivalence groupoid of the subclass associated with the other secondary gauge , which can also be realized after gauging , has even much more involved normalization properties. This is why it is essential to select the gauge of arbitrary elements that is optimal for group classification of the class (1). The careful analysis has resulted in the selection of the initial gauge and the secondary gauge , and this selection of, especially, the secondary gauge is quite unforeseen without having the complete description of the equivalence groupoid of the class (1). Both the subclasses (4) and (5), which are associated with the gauges and , respectively, are normalized in the usual sense. Due to the normalization, we easily control the changes of equivalence groups in the course of gauging. The group classifications of the class (1) and of its subclass (5) are equivalent, cf. Proposition 7. Moreover, the usual equivalence group of the maximally gauged subclass (5) is parameterized by just three functions of the only argument . This fact jointly with the normalization property makes it convenient to solve the group classification problem for the subclass (5) using the algebraic method.
The final ingredient for the efficient solution of the group classification problem was the accurate study of properties of the maximal Lie invariance algebras of equations from the subclass (5). As a result, we have found quite strong restrictions on appropriate subalgebras of the equivalence algebra (more precisely, its projection \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}) of the subclass (5), which has simplified the classification of such subalgebras. In particular, we have proved in Lemma 15 that the dimension of maximal Lie invariance algebras of equations from the subclass (5) is less than or equal to five. We have furthermore introduced three invariant integer parameters , and characterizing the dimensions of relevant subspaces of such algebras and found essential low-dimensional restrictions for the values that the parameters ’s can assume. The conditions for subspaces related to time transformations that are presented in Lemma 18 are highly common for evolution equations and their systems, cf. [3, 19]. In contrast to this, the restriction for values of the pair of parameters that is given in Lemma 17 is specific for the subclass (5) but is still important for effectively classifying appropriate subalgebras of \mathfrak{g}_{\mbox{\tiny\langle,\rangle}}. The restrictions for ’s collected in Section 6 are in total not complete, i.e., there are values of ’s among selected ones that are associated with no maximal Lie invariance algebras of equations from the subclass (5); see Remark 23. At the same time, finding the exhaustive set of restrictions for ’s in fact needs carrying out the complete group classification of the subclass (5). In this sense, the present classification is more similar to the group classification of linear Schrödinger equations in [19] than the group classification of linear evolution equations in [3], where all selected potential values for analogous invariant integer parameters are admitted by some appropriate subalgebras.
Due to the normalization of the class (1) in the usual sense (resp. the subclass (5)), we were able to classify Lie reductions of equations from the class with respect to its usual equivalence group using the technique proposed in [32] and modified in [3]. We have constructed solutions that are at most affine in to equations from a quite wide subclass of the class (1) as well as have discussed generating new solutions from known ones by equivalence transformations.
In spite of the comprehensive symmetry analysis of the class (1) and a number of its subclasses in the present paper, it is pertinent to carry out group classification of other subclasses of the class (1) even for particular small values of . We plan to solve the group classification problem for the class of variable-coefficient Burgers equations of the form with , which was considered in [38] (see a discussion in the introduction of the present paper). Although this class looks simple, its group classification is quite tricky and is supposed to involve partition into subclasses jointly with successively arranging and classifying each part separately.
Although initially the main purpose of the paper had been the exhaustive solution of the group classification problem for the class (1) of general variable-coefficient Burgers–KdV equations of arbitrary fixed order, the study of the equivalence groupoids of subclasses of this class led to more important results, which are worth recalling. For a long time after the first discussion of the notion of generalized equivalence groups in [25, 26], no examples of nontrivial generalized equivalence groups were known in the literature, except classes for which some of arbitrary elements are constants and thus some of components of equivalence transformations associated with system variables depend on such arbitrary elements; see, e.g., [36, Section 6.4], [40, Section 2] and [43, Section 3]. Note that in all these papers, effective generalized equivalence groups were given instead of the corresponding generalized equivalence groups. This is why certain doubts started to circulate in the symmetry community whether this notion is valuable at all. In the present paper we have happened to construct for the first time several examples of nontrivial generalized equivalence groups such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the corresponding classes. All related classes are (reparameterized) subclasses of the class (1). The most significant consequence of the construction of these examples is that they make evident the necessity of introducing the notion of effective generalized equivalence group. Moreover, they also answer, just by their existence, some theoretical questions, which leads to properly posing further questions. In particular, the entire generalized equivalence group of a class may be effective itself and thus it is a unique effective generalized equivalence group of this class, see Remark 11. Nevertheless, there are classes of differential equations admitting multiple effective generalized equivalence groups. As discussed in Remark 28, this claim is exemplified by classes and , for which we have constructed effective generalized equivalence groups that are proper but not normal subgroups of the corresponding generalized equivalence groups. All known examples of generalized equivalence groups that are related to constant arbitrary elements have the same property. Then the natural question is whether there exists a class of differential equations with effective generalized equivalence group being a proper normal subgroup of the corresponding generalized equivalence group. By the way, Corollary 31 shows that even merely singling out an effective generalized equivalence group from the already known generalized equivalence group of a class may be a nontrivial problem.
For the class of general Burgers–KdV equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we have explicitly constructed its extended generalized equivalence group in a rigorous way. We have reparameterized the class introducing virtual arbitrary elements that are nonlocally related to the native arbitrary elements of this class. This is the first example of such a construction for partial differential equations in the literature. Similar results were earlier obtained only for classes of linear ordinary differential equations in the preprint version of [4]. The reparameterization technique developed in the present paper gives hope to us that such construction will be realized soon for many classes of differential equations.
The equivalence groupoid of the class of general Burgers–KdV equations with coefficients depending at most on has an elegant structure but this structure seems uncommon in comparison with various kinds of normalization. Two equations from the class are related by a point transformation that is not the projection of a transformation from the corresponding usual equivalence group if and only if these equations belong to a case of Lie symmetry extension for this class. At the same time, the transformation is in general not decomposed into the projection of an equivalence transformation and a Lie symmetry transformation, and thus the class is even not semi-normalized. Hence no wonder that this example is unique.
Acknowledgments
The authors are grateful to Vyacheslav Boyko, Christodoulos Sophocleous and Olena Vaneeva for useful discussions and interesting comments. This research was undertaken, in part, thanks to funding from the Canada Research Chairs program and the NSERC Discovery Grant program. The research of ROP was supported by the Austrian Science Fund (FWF), project P25064, and by “Project for fostering collaboration in science, research and education” funded by the Moravian-Silesian Region, Czech Republic.
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