Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws
I. Krasil'shchik, A. Sergyeyev
TL;DR
This paper develops an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable PDE related to anti-self-dual vacuum Einstein equations with a nonzero cosmological constant, using a nonisospectral Lax pair.
Contribution
It introduces a novel hierarchy of conservation laws and constructs an infinite-dimensional differential covering for the Przanowski equation, advancing understanding of its integrability properties.
Findings
Established an infinite hierarchy of nonlocal conservation laws.
Constructed a nonisospectral Lax pair for the Przanowski equation.
Derived an infinite-dimensional differential covering.
Abstract
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Integrability of anti-self-dual vacuum Einstein equations
with nonzero cosmological constant:
an infinite hierarchy of nonlocal conservation laws
I. Krasil*′*shchik
V.A. Trapeznikov Institute of Control Sciences RAS, Profsoyuznaya 65, 117342 Moscow, Russia & Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia
and
A. Sergyeyev
Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic
Abstract.
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.
Introduction
Integrable systems play an important role in modern theoretical and mathematical physics, see e.g. [1, 4, 11, 13, 14, 20, 28, 31, 32, 35, 36, 37], and this is particularly true for integrable systems in four independent variables a.k.a. 3+1 dimensions, cf. e.g. [11, 25, 32, 36] and references therein, since according to general relativity our spacetime is four-dimensional. Moreover, a number of integrable (3+1)-dimensional systems of immediate relevance for physics arises within general relativity upon imposition of (anti)self-duality conditions, see, for instance, [1, 3, 25] and references therein, as is the case for the Przanowski equation studied below.
Namely, Przanowski [30] has shown that locally every anti-self-dual Einstein four-manifold admits a compatible complex structure and the metric has the form
[TABLE]
Here is a real function on , and are holomorphic coordinates on and and denote their complex conjugates; is the cosmological constant, cf. e.g. [8] and references therein. As usual, the subscripts indicate partial derivatives, e.g. etc.
The metric (1) is, see [30], an anti-self-dual Einstein metric if and only if satisfies the Przanowski equation
[TABLE]
which is a subject of intense research, see e.g. [2, 12, 21] and references therein.
We leave aside the case of as then, upon having imposed (anti)self-duality on the metric one has to use a normal form of the metric different from (1) and arrives instead of (2) at a different PDE, cf. e.g. [1, 21, 25, 29, 33, 34] and references therein.
Hoegner [12] has established integrability of (2) by constructing a nonisospectral Lax pair for (2) of the form
[TABLE]
where , , and
[TABLE]
Here is an additional independent variable which plays the role of (variable) spectral parameter, see [9, 24, 32, 37] and references therein; we stress that .
The presence of the above Lax pair makes it possible, at least in principle, to obtain exact solutions for (2) using the inverse scattering transform, cf. e.g. [24] and references therein, or the twistorial methods, see, for example, [3, 11, 12, 25] and references therein.
Existence of infinite hierarchies of conservation laws is a well-known feature of integrable systems, cf. e.g. [20, 28] and references therein, and we show below how to construct such a hierarchy for the Przanowski equation using a modification of the above Lax pair. Moreover, we prove that the conservation laws in question are nontrivial and linearly independent.
The rest of the article is organized as follows. In Section 1 we construct an infinite hierarchy of the nonlocal conservation laws in question. In Section 2 we set the stage for Section 3 and our main result, Theorem 1, establishing linear independence and nontriviality for the conservation laws from the hierarchy in question. Section 4 contains discussion.
1. Infinitely many nonlocal conservation laws for the Przanowski
equation
Proposition 1**.**
Equation (2) admits a Lax pair , ,* where and*
[TABLE]
Proof.
It suffices to observe that are related to by the change of variables . ∎
The Lax operators enjoy a simpler structure than . Indeed, they can be written as
[TABLE]
where is defined by the formulas
[TABLE]
i.e., the quantity is a nonlocal variable, namely, a potential for the following local conservation law for (2):
[TABLE]
We stress that the derivatives and are not determined by (4), and hence all quantities for are functionally independent nonlocal variables. In a similar fashion we have infinitely many nonlocal variables in (11) below. This is a fairly common phenomenon for nonlocal variables associated with partial differential systems in more than two independent variables, cf. e.g. Remark 3.9 in [20].
Substituting a formal Taylor expansion into the equations shows that is an arbitrary smooth function of and , and satisfies the equations
[TABLE]
whence
[TABLE]
where is again an arbitrary smooth function of and .
We now see that for , satisfy the recursion relations
[TABLE]
Thus, we have an infinite-dimensional (differential) covering over (2) defined by (7), cf. e.g. [7, 20] for general background on coverings.
System (7) gives rise to an infinite hierarchy of nonlocal conservation laws for (2):
[TABLE]
where , , , , , , , , and and are defined by the following relations:
[TABLE]
and for , , we set
[TABLE]
The conservation laws (8) are linearly independent and nontrivial, as we are going to establish in Theorem 1 below.
In closing note that (2) enjoys an obvious discrete symmetry , which however does not extend to its Lax operators or . This implies that there exists another infinite hierarchy of nonlocal conservation laws for (2) obtained from (8) by the simultaneous swap , .
2. Nontriviality of conservation laws: preliminaries
2.1. Simplifications
Before proceeding further, notice that the problem under study admits some useful simplifications. Namely, note that
- (1)
Under the rescaling equation (2) transforms into
[TABLE]
and thus we can set in all subsequent computations without loss of generality. 2. (2)
Coverings (6) and (4) are equivalent in the sense of [19], which inter alia implies that we can set without loss of generality , .
Thus, the infinite-dimensional covering defined by (6) and (7) boils down to
[TABLE]
and
[TABLE]
for .
2.2. Coordinates and total derivatives
We rewrite (2), where we set as per Section 2.1, in the form
[TABLE]
and choose internal coordinates on the associated diffiety , i.e., the infinite prolongation of (9), as follows (see e.g. [7] for the background on geometry of diffieties):
[TABLE]
where , , .
Then the total derivatives on read
[TABLE]
where
[TABLE]
is the right-hand side of (9).
To introduce nonlocal variables, we, for convenience of notation, do some relabeling, namely, let and , . Then the nonlocal variables employed below are
[TABLE]
The total derivatives lifted to the covering equation are
[TABLE]
where the nonlocal tails
[TABLE]
are defined by the formulas
[TABLE]
and , are as defined above and take the form
[TABLE]
here and
[TABLE]
while
[TABLE]
where , are the right-hand sides of (10).
In what follows we shall need the following presentation of the coefficients :
[TABLE]
where and denotes the terms of lower jet order both in and which are inessential for the subsequent computations.
3. Nontriviality of conservation laws: the proof
Equations (10)–(11) define an infinite family of (nonlocal) conservation laws
[TABLE]
for equation (9).
In other words, on we have
[TABLE]
where and .
Note that all these conservation laws are two-component in the sense that expressions (14) involve only two total derivatives, and , out of four.
It could be of interest to find out whether (9) also has three- or four-component conservation laws, cf. e.g. [22, 23] and references therein, and to explore nonlocal symmetries for (9) involving nonlocal variables being potentials for the conservation laws (13). Note that our computations show that there are no three- or four-component local conservation laws of order up to four for (9), and we strongly suspect that none exist even if we proceed to higher orders, cf. the discussion at the end of Section 4. However, the matter of existence of nonlocal three- or four-component conservation laws for (9) remains an interesting open problem.
We now intend to prove that the system of the conservation laws under study is nontrivial. Let us clarify this claim.
The system of conservation laws defines the tower of coverings
[TABLE]
where is the infinite prolongation of the Przanowski equation, while the covering equations contain the nonlocal variables , , with being the inverse limit. In a similar fashion, we define by induction the towers
[TABLE]
and
[TABLE]
We are going to prove the following
Theorem 1**.**
For any ,* an arbitrary finite system of conservation laws of the equation is linearly independent.*
The nontriviality of these conservation laws is, in view of their structure, see (8), a straightforward consequence of their linear independence.
The proof of Theorem 1 will be based on the following
Proposition 2** (see [16], cf. also [17, 18]).**
Let be a differentially connected equation111Recall that an equation is called differentially connected if the only functions that are invariant with respect to all total derivatives on are constants.. The conservation laws are mutually independent in the sense of Theorem 1 if and only if is differentially connected as well,* i.e., the only solutions of the system*
[TABLE]
are constants.
Proof of Theorem 1.
We begin the proof with an obvious observation that the Przanowski equation (9) is differentially connected. Before proceeding further with the proof of the theorem, let us briefly describe the outline of the former. Namely, we will prove that the space consists of functions that depend on , , and only, from where the desired result immediately follows. To this end, we will perform double induction: on in (16) and on in (15) for each . The case of is special and is considered separately. So, the base of induction is .
Let us employ the notation for the space of functions that belong to and depend on a finite set of internal coordinates in and of nonlocal variables , , .
Lemma 1**.**
If then .
Proof of Lemma 1.
The desired result is a straightforward consequence of the defining equations (10) and (11). ∎
Let us now pass to the proof of the theorem.
Step** 1**** ().**
We prove here by induction on that the conservation laws are linearly independent. Denote and perform induction on .
First, let . Then
[TABLE]
This means that the set may consist of the variables , , , , and only, i.e.,
[TABLE]
and consequently
[TABLE]
Thus, .
Next, let . Then
[TABLE]
In this expession, the variables of the maximal jet order are , . Consequently, is invariant with respect to the vector fields
[TABLE]
Now notice that the coefficients of at are independent of the variables , . From this fact it immediately follows that cannot depend on , . Hence, taking commutators of the vector fields with we obtain that is invariant with respect to the derivations , and this completes the induction step.
Step** 2**** **(, the base of induction
on ).
Let now .
Lemma 2**.**
.
Proof of Lemma 2.
Consider the total derivative
[TABLE]
Assume that . Then differentiating with respect to using (17) we find that is invariant with respect to the vector field
[TABLE]
Consequently, it is invariant with respect to the commutator
[TABLE]
i.e., does not depend on for . ∎
Set ; then and let . The internal coordinates of maximal order on which the coefficients at and in depend are
[TABLE]
Differentiating with respect to these coordinates, we see that must be invariant with respect to the vector fields
[TABLE]
Note that .
We now use the induction on . The base is . Compute the commutator
[TABLE]
Hence, is invariant with respect to
[TABLE]
Next,
[TABLE]
and thus the vector field
[TABLE]
annihilates as well. But
[TABLE]
which means that does not depend on , i.e., we find ourselves in the situation of Step 1.
Now pass to the induction step and assume . Just as in the preceding computations, we see that the commutator is proportional to the vector field
[TABLE]
while the commutator is proportional to
[TABLE]
Thus, is independent of , so we change the field to
[TABLE]
Then is proportional to the vector field
[TABLE]
while the commutator equals, up to a factor, to the vector field
[TABLE]
Hence, is independent of and instead of we can take the vector field
[TABLE]
etc. Eventually, we shall arrive at the independence of on all the variables , and this completes the induction step for .
Step** 3**** (the induction step).**
Let
Lemma 3**.**
One has ,* , i.e., , where .*
Proof of Lemma 3.
The proof is similar to that of Lemma 2. ∎
Thus, , and the internal coordinates of maximal jet order on which the coefficients in the total derivatives at that act nontrivially on may depend are
[TABLE]
Therefore, is invariant with respect to the vector fields
[TABLE]
We first fix and set (the base of induction). Then
[TABLE]
Hence the commutator is proportional to
[TABLE]
for the commutator we get
[TABLE]
etc., and equals, up to a functional factor, to the vector field
[TABLE]
Finally, for we get
[TABLE]
Therefore,
[TABLE]
and does not depend on .
Let now . Applying a similar procedure of step-by-step commutation with , we find that we have
[TABLE]
where denotes proportionality. In particular,
[TABLE]
Then we have
[TABLE]
This leads to the invariance of with respect to the fields and . In addition, using equations (19) we obtain by induction independence of on all the variables . In particular, this means that instead of in equations (18) we can consider the vector field
[TABLE]
Using (12) we get
[TABLE]
But, thanks to the previous remark, the last summand can be omitted and we can set
[TABLE]
Proceeding in a similar fashion as above, we shall obtain the vector fields
[TABLE]
In particular,
[TABLE]
and
[TABLE]
Using the same reasoning as for the fields , we deduce that does not depend on the variables , etc. Eventually, we shall arrive at the independence of of all variables . This completes the induction step.
Theorem 1 is proved. ∎
4. Closing remarks
In the present paper we have found infinitely many nonlocal conservation laws for the Przanowski equation and established their nontriviality.
To put these results into a context, recall that employing isospectral Lax pairs for the generation of conservation laws dates back to 1968, see [26] where an infinite hierarchy of local conservation laws for the celebrated Korteweg–de Vries equation was constructed. However, as we have seen above, nonisospectral Lax pairs can be successfully employed for the same purpose too. A similar procedure was applied to other equations in [5, 17, 18, 32] and can be canonically associated to any differential covering in a geometrical framework, see [15]. On the other hand, while isospectral Lax pairs can also be applied for the construction of recursion operators in a fairly straightforward fashion, cf. e.g. [6, 11, 13, 27, 31] and references therein, to the best of our knowledge this does not seem to be the case for nonisospectral Lax pairs, especially in the case of more than two independent variables.
Note that the proof of nontriviality of the conservation laws obtained from a nonisospectral Lax pair can be quite hard, see the proof of Theorem 1 above, and, moreover, special handling of each particular equation could be required, cf. e.g. [5, 17, 18]. The conservation laws resulting from the procedure in question are usually nonlocal, and it is natural to ask which is the use and meaning thereof.
Let be a partial differential system in dependent variables and independent variables , and be a covering over this system, cf. e.g. [7, 20] for details. Then a -nonlocal conservation law of is nothing but a “usual”, i.e., local, conservation law of the covering system . Thus, a -nonlocal conservation law for is a closed -differential form on whose coefficients can depend not just on the quantities , but, informally speaking, also on the quantities like , where
[TABLE]
are the equations that define the nonlocal variables in the covering . So, in many respects, nonlocal conservation laws are quite similar to local ones, and share a number of possible applications with the latter.
For instance, for an evolutionary system nonlocal conservation laws yield constants of motion in essentially the same way as local ones. If, moreover, the evolutionary system under study admits a Hamiltonian structure, one can, under certain conditions, employ this structure to obtain (in general, nonlocal) symmetries from the nonlocal conserved densities in the same fashion as for the densities of local conservation laws, cf. e.g. [28]. On the qualitative level, the presence of an infinite hierarchy of nonlocal conservation laws, just as for that of their local counterparts, can be seen as indicative of integrability as it imposes strong constraints on the associated dynamics making the latter highly regular, cf. e.g. [1, 11] and references therein.
In closing note that our computations revealed only two local conservation laws of the Przanowski equation (2): given by formulas (4) and its “twin” arising as a result of the discrete symmetry mentioned at the end of Section 1, that is, , . Symbolic computations show that there exist no other local conservation laws of the order up to four. Unfortunately we have no rigorous nonexistence proof for arbitrarily high order, and in fact it would be quite interesting to find such a proof.
Acknowledgments
The work of IK was partially supported by the RFBR Grant 18-29-10013 and IUM-Simons Foundation. The research of AS was supported in part by the Ministry of Education, Youth and Sport of the Czech Republic (MŠMT ČR) under RVO funding for IČ47813059 and the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028. AS is pleased to thank A. Borowiec, M. Dunajski and K. Krasnov for stimulating discussions. AS also warmly thanks the Institute of Theoretical Physics of the University of Wrocław, and especially A. Borowiec, for the warm hospitality extended to him in the course of his visits to Wrocław, where some parts of the present article were worked on.
It is our great pleasure to thank the anonymous referee for useful and relevant comments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering , Cambridge University Press, Cambridge, 1991.
- 2[2] S. Alexandrov, B. Pioline, and S. Vandoren, Self-dual Einstein spaces, heavenly metrics, and twistors, J. Math. Phys. 51 (2010), 073510, ar Xiv:0912.3406
- 3[3] M. Atiyah, M. Dunajski, L.J. Mason, Twistor theory at fifty: from contour integrals to twistor strings, Proc. R. Soc. A 473 (2017), no. 2206, art. 20170530
- 4[4] O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems , Cambridge University Press, Cambridge, 2003.
- 5[5] H. Baran, P. Blaschke, I.S. Krasil ′ shchik, M. Marvan, On symmetries of the Gibbons–Tsarev equation, J. Geom. Phys., to appear, https://doi.org/10.1016/j.geomphys.2019.05.011, ar Xiv:1811.08199
- 6[6] H. Baran, I.S. Krasil ′ shchik, O.I. Morozov, P. Vojčák, Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study, Theor. Math. Phys. 196 (2018), no. 2, 1089–1110, ar Xiv:1611.04938.
- 7[7] A.V. Bocharov et al., Symmetries of Differential Equations in Mathematical Physics and Natural Sciences , edited by A.M. Vinogradov and I.S. Krasil ′ shchik). Factorial Publ. House, 1997 (in Russian). English translation: Amer. Math. Soc., 1999.
- 8[8] P. Bull et al., Beyond Λ Λ \Lambda CDM: Problems, solutions, and the road ahead, Physics of the Dark Universe 12 (2016), 56–99, ar Xiv:1512.05356