Reduction of lattice equations to the Painlev\'e equations: P$_{\rm IV}$ and P$_{\rm V}$
Nobutaka Nakazono
TL;DR
This paper establishes a new link between ABS lattice equations and Painlevé equations, enabling the construction of difference-differential Lax representations for P_IV and P_V, advancing the understanding of integrable systems.
Contribution
It introduces a novel relation between ABS equations and Painlevé equations, and constructs their difference-differential Lax representations.
Findings
New relation between ABS and Painlevé equations
Construction of difference-differential Lax representations for P_IV and P_V
Enhanced understanding of integrable lattice systems
Abstract
In this paper, we construct a new relation between ABS equations and Painlev\'e equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlev\'e equations.
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Reduction of lattice equations to the Painlevé equations: PIV and PV
Nobutaka Nakazono
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan.
Abstract.
In this paper, we construct a new relation between ABS equations and Painlevé equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlevé equations.
Key words and phrases:
Painlevé equation; ABS equation; Lax representation; Hamiltonian; affine Weyl group
2010 Mathematics Subject Classification:
33E17,37K05,37K10,34M55,34M56,39A14
1. Introduction
In recent works by Joshi-Nakazono-Shi [14, 13, 16, 15], the mathematical connection between two longstanding classifications of integrable systems in different dimensions, one by Adler-Bobenko-Suris (ABS equations) [2, 10, 3, 5, 6, 7] and the other by Okamoto and Sakai (Painlevé and discrete Painlevé equations) [35, 27], have been investigated by using their lattice structures. Moreover, a comprehensive method of constructing Lax representations of discrete Painlevé equations using this connection was provided in [12] and demonstrated in [12, 15]. The whole picture of the connection between the ABS equations and the discrete Painlevé equations has been gradually revealed, but that between the ABS equations and the (differential) Painlevé equations was missing, that is, there is still a great distance between them. In the present paper, we fill this gap by erecting a bridge from the ABS equations to the Painlevé equations.
A hierarchy of nonlinear ordinary differential equations (ODEs) found by Noumi-Yamada in [24, 22] is sometimes referred to as NY-system. It is well known that NY-system contains the fourth and fifth Painlevé equations (PIV and PV) and has an -type affine Weyl group symmetry.
In this paper, we show that a system of ABS equations can be reduced to NY-system by a periodic type reduction. Through this connection we construct the difference-differential Lax representations of PIV and PV (see Theorems 1.3 and 1.5). Moreover, we obtain remarkable results that the dependent variable of the system of ABS equations can be reduced to the Hamiltonians of PIV and PV (see Theorems 1.2 and 1.4).
1.1. The fourth and fifth Painlevé equations
In this paper, we focus on the following Painlevé equations:
[TABLE]
where
[TABLE]
and
[TABLE]
where
[TABLE]
Note that in both cases, are dependent variables, is the independent variable, are complex parameters and ′ denotes . The polynomial Hamiltonians of PIV and PV [28, 29] are respectively given by and , where
[TABLE]
Remark 1.1**.**
PIV* (1.1) can be rewritten as the following “standard” symmetric form given in [25, 23, 22]*:**
[TABLE]
by the following replacements:
[TABLE]
Also, we can express PV (1.3) in the following “standard” symmetric form given in [22, 23]:**
[TABLE]
by the following replacements:
[TABLE]
Moreover, by using the replacements (1.8) and (1.10), Equations (1.5) and (LABEL:eqn:intro_hamiltonian_P5) can be rewritten as
[TABLE]
which are the Hamiltonians of Equations (1.7) and (1.9), respectively.
1.2. Main results
In this section, we outline four main results of this paper.
Firstly, in §4.1 we prove the following theorems for PIV (1.1).
Theorem 1.2**.**
The dependent variable of the system of ABS equations (2.1) with can be reduced to the Hamiltonian (1.5).
Theorem 1.3**.**
The Lax representation of PIV (1.1) is given by the following:
[TABLE]
that is, the compatibility condition
[TABLE]
is equivalent to PIV (1.1). Here,
[TABLE]
where the variables and are given by
[TABLE]
and is an arbitrary complex constant.
Other two main results are for PV (1.3) given by the following theorems.
Theorem 1.4**.**
The dependent variable of the system of ABS equations (2.1) with can be reduced to the Hamiltonian (LABEL:eqn:intro_hamiltonian_P5).
Theorem 1.5**.**
The Lax representation of PV (1.3) is given by
[TABLE]
that is, the compatibility condition
[TABLE]
is equivalent to PV (1.3). Here,
[TABLE]
where
[TABLE]
and is an arbitrary complex constant.
The proofs of Theorems 1.4 and 1.5 are given in §4.2.
1.3. Background
The six Painlevé equations: PVI, …, PI are nonlinear ODEs of second order which have the Painlevé property, i.e., their solutions do not have movable branch points. It is well known that the Painlevé equations, except for PI, have Bäcklund transformations, which collectively form affine Weyl groups. The following is the diagram of degenerations:
[TABLE]
where the symbols inside the parentheses indicate the types of affine Weyl groups. (See [26, 33, 32, 31, 30, 22, 35, 17] for the details.)
In [2, 10, 3, 5, 6, 7], Adler-Bobenko-Suris (ABS) et al. classified polynomials , say, of four variables into 11 types: Q4, Q3, Q2, Q1, H3, H2, H1, D4, D3, D2, D1. The first four types, the next three types and the last four types are collectively called -, - and -types, respectively. The resulting polynomial satisfies the following properties.
**(1) Linearity: **
is linear in each argument, i.e., it has the following form:
[TABLE]
where coefficients are complex parameters.
**(2) 3D consistency and tetrahedron property: **
There exist a further seven polynomials of four variables: , , which satisfy property (1) and a cube on whose six faces the following equations are assigned:
[TABLE]
where the eight variables , …, lie on the vertices of the cube (see Figure 1), in such a way that can be uniquely expressed in terms of the four variables , , , (3D consistency) and moreover the following relations hold (tetrahedron property):
[TABLE]
Since these equations relate the vertices of the quadrilateral on a lattice, they are often called quad-equations or lattice equations.
Some polynomials of ABS type are
[TABLE]
where and . Many well known integrable partial difference equations (PEs) arise from assigning a polynomial of ABS type to quadrilaterals in the integer lattice , for example:
**discrete Schwarzian KdV equation[18, 20]: **
[TABLE]
**lattice modified KdV equation[18, 21, 2]: **
[TABLE]
**lattice potential KdV equation[11, 18]: **
[TABLE]
where
[TABLE]
Throughout this paper, we refer to such PEs as ABS equations.
We note that in general a hypercube is said to be multi-dimensionally consistent, if all cubes contained in the hypercube are 3D consistent (see property (2) above). In a similar manner to the construction of ABS equations a hypercube causes a system of ABS equations by tilling it in the integer lattice (see, for example §2).
It is well known that the Painlevé equations arise as the monodromy-preserving deformation of linear differential equations (see e.g., [36, 17, 22, 8, 9] and reference therein). The pair of linear differential equation and its deformation equation is referred to as the Lax representation (or, Lax pair) of the corresponding Painlevé equation. It has also been reported that a compatibility condition of linear difference equation and its deformation equation also give a Painlevé equation [17, 34, 1]. We here denote such a Lax representation as a difference-differential Lax representation. Lax representations of the Painlevé equations usually arise by reductions from the integrable partial differential equations such as KdV equation, modified KdV equation, and so on. In this paper, we show that difference-differential Lax representations of the Painlevé equations can be obtained from a system of integrable PEs of ABS type through periodic type reductions by using PIV and PV as examples. Note that a Lax representation of an ABS equation is given by a pair of linear difference equation and its spectrum-preserving deformation. For a relation between monodromy- and spectrum- preserving deformations, we refer to [8].
1.4. Plan of the paper
This paper is organized as follows. In §2, we first define the system of ABS equations (2.1) and construct its Lax representation. Then, we consider the reduction of system (2.1) to the system of ODEs (2.19). In §3, using the symmetry of the integer lattice we obtain the affine Weyl group symmetry and the difference-differential Lax representation of system (2.19). In §4, considering the relation between system (2.19) and NY-system, we give the proofs of Theorems 1.2–1.5. Some concluding remarks are given in §5.
2. Reduction of a system of ABS equations to a system of ODEs
In this section, we consider the periodic reduction of the system of ABS equations (2.1) to the system of ODEs (2.19), which is equivalent to NY-system (see §4 for the details).
In the same way that the lattice can be constructed by tiling the plane with squares, we construct the lattice , where , by tiling it with -dimensional hypercubes (i.e. -cubes). We obtain a system of PEs on the lattice in a similar manner to the constructions of the ABS equations (see §1.3). Indeed, assigning the function and H1ϵ=0 equations to the vertices and faces of each -cube, we obtain the following system of ABS equations:
[TABLE]
where is the dependent variable and
[TABLE]
are complex parameters. Here, the subscript (or, ) for an arbitrary function means shift (or, shift) in the -direction, that is,
[TABLE]
Below, we also use these notations for other objects. For example, (2.1) denotes (2.1).
We first rewrite system (2.1) by perceiving the -direction as special. Let and be the functions of as follows:
[TABLE]
where . Then, system (2.1) can be rewritten as
[TABLE]
where . Here, the overline for an arbitrary function means shift of , that is,
[TABLE]
Note that system (2.4) is not a special case of system (2.1). Indeed, shifting to and using replacement (2.3), we inversely obtain system (2.1) from system (2.4).
Following the method given in [4, 19, 38, 12], we obtain the Lax representation of system (2.4).
Lemma 2.1**.**
The Lax representation of system (2.4) is given by
[TABLE]
where , and is the spectral parameter, that is, the compatibility conditions
[TABLE]
are equivalent to (2.4a) and (2.4b), respectively.
In Appendix A, we give the proof of Lemma 2.1 and show how to construct a Lax representation of a system of ABS equations by using system (2.4) as an example.
We next consider a periodic reduction of system (2.4) and its Lax representation (2.6).
Lemma 2.2**.**
Let
[TABLE]
where
[TABLE]
By imposing the -periodic condition
[TABLE]
with the following conditions of the parameters for l\in\mathbb{Z}$$:
[TABLE]
system (2.4) is reduced to the following system of equations:
[TABLE]
where and ′ denotes . Moreover, the Lax representation (2.6) is also reduced to the Lax representation of system (2.12) given by
[TABLE]
where and , that is, the compatibility conditions
[TABLE]
are equivalent to (2.12a) and (2.12b), respectively.
The proof of Lemma 2.2 is given in Appendix B.
We are now in a position to get a system of ODEs. Let us define the variables and the parameters , , by
[TABLE]
where
[TABLE]
Substituting
[TABLE]
into system (2.12a) and using the relation
[TABLE]
which can be verified by the direct calculation, we obtain the following system of ODEs:
[TABLE]
where . System (2.19) is equivalent to NY-system (see §4 for the details). Before explaining it, in the next section we consider the affine Weyl group symmetry and Lax representation of system (2.19).
3. Affine Weyl group Symmetry and Lax representation of system (2.19)
In this section, we first consider a linear action of an affine Weyl group of -type, which is the symmetry of the integer lattice. Then, lifting its action to a birational action we obtain affine Weyl group symmetry of system (2.19). Using the symmetry group together with Lemma 2.1, we finally construct the difference-differential Lax representation of system (2.19).
3.1. Affine Weyl group Symmetry of the lattice
In this section, considering a symmetry of the integer lattice , we obtain the linear action of the affine Weyl group .
We define the automorphisms of the lattice : , , and by the following actions on the coordinates :
[TABLE]
For convenience, throughout this paper we use the following notation for the combined transformation of arbitrary mappings and :
[TABLE]
The group of automorphisms forms the extended affine Weyl group of type , denoted by . Namely, they satisfy the following fundamental relations:
[TABLE]
where . Note that is not the “full” extended affine Weyl group of type , since it only includes rotational symmetries of the affine Dynkin diagram, but not the reflections.
Remark 3.1**.**
Action of each element of on the coordinates is defined, but that on the each lattice parameter is not defined. For example, in the case the transformation acts on the origin as the following (see Figure 2):
[TABLE]
but it cannot act on the parameter like .
In the lattice , there are orthogonal directions, which naturally give rise to translation operators. Operators , , whose actions on the coordinates are given by
[TABLE]
can be expressed by the elements of as the following:
[TABLE]
where . Note that is also a translation operator whose action is given by
[TABLE]
and can be expressed by compositions of as the following:
[TABLE]
3.2. Affine Weyl group Symmetry of system (2.19)
In this section, extending the linear action of given in §3.1 to the birational action, we obtain the affine Weyl group symmetry of system (2.19).
From the definition, the variables , defined by (2.8a), are assigned on the vertices and the quad-equations (2.12b) are assigned on the faces of the lattice . Therefore, we can naturally lift the action of to the actions on the function by
[TABLE]
where . These actions give the actions on the parameters and the variables as the following lemma.
Lemma 3.2**.**
The actions of on the parameters , , are given by
[TABLE]
where , while those on the variables , , are given by
[TABLE]
where .
Proof.
From the periodic condition (2.10), the definition (2.15) and the actions (3.9), we obtain
[TABLE]
Moreover, substituting
[TABLE]
into (2.12b), we obtain
[TABLE]
where , respectively. Therefore, from Equations (3.12), (3.13), (3.16) and (3.17), the actions (3.11a) hold. From the actions (3.9) and the definition (2.15), the others can be easily verified. Therefore, we have completed the proof. ∎
In general, for a function , we let an element act as , that is, acts on the arguments from the left. We can easily verify that under the birational actions given in Lemma 3.2, satisfies the fundamental relations (3.3) and the following relation:
[TABLE]
We can also verify that by using system (2.19) and the birational actions of given in Lemma 3.2, the following relations hold:
[TABLE]
where , which indicate that is a Bäcklund transformation group of system (2.19). Therefore, the following lemma holds.
Lemma 3.3**.**
Bäcklund transformations of system (2.19) collectively form the extended affine Weyl group .
3.3. Difference-differential Lax representation of system (2.19)
In this section, we define the birational action of on the wave function . Using this action, we obtain the difference-differential Lax representation of system (2.19).
In a similar manner to , we can assign on the vertices . Then, the action of can be lifted to the action on the function as the following:
[TABLE]
where
[TABLE]
Let us define the variables , , and the parameter by
[TABLE]
Substituting
[TABLE]
into system (2.13), we obtain the following system of equations:
[TABLE]
where and , , and are defined by
[TABLE]
Then, the action of on the variables , , and the parameter are given in the following lemma.
Lemma 3.4**.**
The action of on the parameter is given by
[TABLE]
while that on the variables , , is given by
[TABLE]
where and .
Proof.
From (3.9), (3.20), (3.24b) and (3.22), we obtain
[TABLE]
[TABLE]
[TABLE]
Therefore, from (3.28), (3.29) and (3.30), we obtain (3.27a), (3.27b) and (3.27d), respectively. From the actions (3.1) and (3.20) and the definition (3.22), the others can be easily verified. Therefore, we have completed the proof. ∎
Note that we can easily verify that under the actions on the variables and the parameter , satisfies the fundamental relations (3.3) but does not satisfy the relation (3.18). This unsatisfied relation is a key to construct a difference-differential Lax representation of an ODE.
We are now in a position to construct a Lax representation of system (2.19). Let us define the shift operator of by
[TABLE]
The action of on the parameter is given by
[TABLE]
while that on the variables and parameters , , is given by an identity mapping, i.e.,
[TABLE]
Therefore, from (3.24a) and (3.27d) we obtain the following lemma.
Lemma 3.5**.**
The difference-differential Lax representation of system (2.19) is given by the following:
[TABLE]
where
[TABLE]
that is, the compatibility conditions
[TABLE]
are equivalent to system (2.19). Note that the relations between the parameters , and the parameters , are given by (3.25).
4. Difference-differential Lax representations of PIV and PV
In this section, considering the relation between system (2.19) and NY-system, we obtain the difference-differential Lax representations of PIV (1.1) and PV (1.3).
Let us define the variables , , by
[TABLE]
where . Then, system (2.19) can be rewritten as the periodic dressing chain with period [37]:
[TABLE]
where . It is well known that from system (4.2), we can obtain NY-system containing PIV and PV[36, 1]. Through this relation we construct the Lax representations of PIV and PV from the Lax representation of system (2.19).
4.1. Case : the Painlevé IV equation
In this section considering the case we obtain the difference-differential Lax representation of PIV from that of system (2.19).
Let us define the variables , , by
[TABLE]
Then, from system (2.19) and the condition for the parameters (2.16) with , we obtain PIV (1.1) with the conditions (1.2).
From the affine Weyl group symmetry of system (2.19) given in Lemma 3.2, we obtain that of PIV as follows. The actions of on the parameters , , are given by
[TABLE]
where , while those on the variables , , are given by
[TABLE]
where . Under these actions, the fundamental relations for hold:
[TABLE]
where . The corresponding Dynkin diagram is given by Figure 3. Before the discussion of the Lax representation of PIV, let us consider the role of -variables in the theory of the Painlevé IV equation.
Lemma 4.1**.**
The following relation holds:
[TABLE]
where is the Hamiltonian given by (1.5).
Proof.
Let
[TABLE]
We can easily verify the following relations:
[TABLE]
Moreover, using the relation
[TABLE]
we obtain
[TABLE]
Equations (4.9) and (4.11) mean that is an arbitrary constant. It is obvious that without loss of generality we can put . Therefore, we have completed the proof. ∎
Therefore, Theorems 1.2 and 1.3 follow from Lemma 3.5 with
[TABLE]
and Lemma 4.1.
4.2. Case : the Painlevé V equation
In a similar manner to the case (see §4.1), the case gives PV (1.3) and its difference-differential Lax representation.
Let
[TABLE]
Then, we obtain PV (1.3) and the conditions (1.4) from system (2.19) and the condition (2.16). The action of extend affine Weyl group on the parameters , , are given by
[TABLE]
where , while those on the variables , , are given by
[TABLE]
where . Under these actions, satisfies the following fundamental relations:
[TABLE]
where . The Dynkin diagram for is given by Figure 4. In a similar manner to the proof of Lemma 4.1, we can prove the following lemma.
Lemma 4.2**.**
The following relation holds:
[TABLE]
where is the Hamiltonian given by (LABEL:eqn:intro_hamiltonian_P5).
Therefore, Theorems 1.4 and 1.5 follow from Lemma 3.5 with
[TABLE]
and Lemma 4.2.
5. Concluding remarks
In this paper, we have constructed the relation between the ABS equations and NY-system through the periodic type reduction. Using this connection, we obtained the difference-differential Lax representations of PIV and PV. Moreover, we showed that the dependent variable of the system of ABS equations (2.1) can be reduced to the Hamiltonians of PIV and PV.
An interesting future project is to investigate the relations between ABS equations and the other Painlevé equations (i.e., PVI, PIII, PII, PI). The results in this direction will be reported in forthcoming publications.
Acknowledgment
The author would like to express his sincere thanks to Profs M. Noumi and Y. Yamada for inspiring and fruitful discussions. I also appreciate the valuable comments from the referee which have improved the quality of this paper. This research was supported by a grant # DP160101728 from the Australian Research Council and JSPS KAKENHI Grant Number JP17J00092.
Appendix A Proof of Lemma 2.1
In this section, we construct the Lax representation of system (2.1) following the method given in [4, 19, 38, 12].
The key to constructing the Lax representation of system (2.1) is to introduce a virtual direction from the lattice , where system (2.1) is assigned, to the multi-dimensionally consistent integer lattice . Then, system (2.1) can be extended to the following system of PEs:
[TABLE]
where . Here, is the dependent variable of system (2.1). Distinguish from . Then, each of equations between and :
[TABLE]
where , can be regarded as the first order discrete system of Riccati type of the quantity , which is linearizable. Indeed, substituting
[TABLE]
in (A.2) and dividing them into the numerators and the denominators, we obtain the following linear systems:
[TABLE]
where the vector is defined by
[TABLE]
We can easily verify that the compatibility conditions
[TABLE]
are equivalent to system (2.1). Finally, using the replacements (2.3) and
[TABLE]
we have completed the proof of Lemma 2.1.
Appendix B Proof of Lemma 2.2
In this section, we give a proof of Lemma 2.2.
By using (2.8), Equations (2.4a) and (2.4b) can be respectively rewritten as
[TABLE]
while Equations (2.6a) and (2.6b) can be respectively rewritten as
[TABLE]
where and . The periodic condition (2.10) imposes the condition that (B.1) is equivalent to (B.1). From the condition that (LABEL:eqn:lpKdVs_U_2) is equivalent to (LABEL:eqn:lpKdVs_U_2), we obtain
[TABLE]
which hold under the conditions of parameters (2.11). From the condition that (LABEL:eqn:lpKdVs_U_1) is equivalent to (LABEL:eqn:lpKdVs_U_1), we obtain the following condition:
[TABLE]
which causes the continuum limit of systems (B.1) and (B.2) to systems (2.12) and (2.13), respectively. Therefore, we have completed the proof of Lemma 2.2.
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