# Nonlocal symmetries of Plebański’s second heavenly equation

- DOI
- 10.1080/14029251.2018.1452669How to use a DOI?
- Keywords
- Plebański’s second heavenly equation; Lax pair; differential covering; nonlocal symmetries
- Abstract
We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering. Also, we find two new infinite hierarchies of commuting nonlocal symmetries in this covering and describe the structure of the Lie algebra of the obtained nonlocal symmetries.

- Copyright
- © 2018 The Authors. Published by Atlantis and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

In this paper we study nonlocal symmetries of Plebański’s second heavenly equation, [32],

*λ*. Expanding the pseudopotential

*q*into a Taylor series

*q*over Eq. (1.1). The goal of this paper is to study nonlocal symmetries for Eq. (1.1) in this covering.

_{k}Infinite-dimensional Lie algebras of nonlocal symmetries are well known to play an important role in the theory of nonlinear integrable equations and provide a useful tool to study of the latter, see e.g. [5,7,16,17,20,33,34] and references therein. Eq. (1.1) has the infinite-dimensional Lie algebra 𝔰 of local contact symmetries and, as it was shown in [24], is uniquely defined by this algebra, see also [11,25] for discussion of geometric properties of Eq. (1.1) and related equations. The algebra 𝔰 is the semi-direct product

^{a}𝔰 ⊕ 𝔞 ⊕ 𝔟 (the sum as vector spaces). In paricular, we show that

There is a great many of works devoted to methods of studying nonlinear partial differential equations (PDEs) that admit nonlocal symmetries. For the case of potential symmetries, that is nonlocal symmetries corresponding to Abelian coverings ^{b}, see [8, Ch.7],[9], and references therein. In regard to applications of nonlocal symmetries in non-Abelian coverings to studying of exact solutions and other integrability properties of nonlinear PDEs, see [5,16,17,34] and references therein.

## 2. Preliminaries

All considerations in this paper are local. The presentation in this section closely follows [20,21], see also [22,23]. Let *J*^{∞}(*π*) be the bundle of its jets of the infinite order. The local coordinates on *J*^{∞}(*π*) are *I* = (*i*_{1},…,*i _{n}*) is a multi-index, and for every local section

*π*the corresponding infinite jet

*j*

_{∞}(

*f*) is a section

*n*= 4,

*m*= 1 we denote

*x*

^{1}=

*t*,

*x*

^{2}=

*x*,

*x*

^{3}=

*y*,

*x*

^{4}=

*z*and

*i*times

*t*,

*j*times

*x*,

*k*times

*y*, and

*l*times

*z*

The vector fields

*i*

_{1},…,

*i*,…,

_{k}*i*) + 1

_{n}*= (*

_{k}*i*

_{1},…,

*i*+ 1, …,

_{k}*i*), are called

_{n}*total derivatives*. They commute everywhere on

The *evolutionary derivation* associated to an arbitrary smooth function

A system of PDEs *s* ≥ 1 with *R* ≥ 1 defines the submanifold *J*^{∞}(*π*).

A function *generator of an infinitesimal*) symmetry of **E**_{φ}(*F*) = 0 on *φ* is a solution to the *defining system*

*Lie algebra*

*of*(

*infinitesimal*)

*symmetries of equation*

*Jacobi bracket*{

*φ, ψ*} =

**E**

*(*

_{φ}*ψ*) −

**E**

*(*

_{ψ}*φ*). The subalgebra of

*contact symmetries*of

Denote *differential covering* of *with the extended total derivatives*

*i*≠

*j*whenever

*w*by

^{s}*covering equations*

A covering is said to be Abelian when system (2.4) can be mapped to the form

Denote by *A shadow of nonlocal symmetry of * corresponding to the covering

*τ*with the extended total derivatives (2.3), or

*τ*-

*shadow*, is a function

*D*(

_{K}*F*) = 0 and (2.4).

*A nonlocal symmetry of*

*τ*(or

*τ*-symmetry) is the vector field

*φ*satisfies (2.5) and

## 3. Local symmetries

The local contact symmetries of Eq. (1.1) are solutions *φ* = *φ*(*t*, *x*, *y*, *z*, *u*, *u _{t}*,

*u*,

_{x}*u*,

_{y}*u*) to the equation

_{z}^{c}show that the Lie algebra

*A*=

*A*(

*t*,

*z*) and

*B*=

*B*(

*t*,

*z*) below are arbitrary functions of

*t*and

*z*. The structure of

### Remark 3.1.

We have *φ _{i}*(

*A*) and

*ψ*The algebra 𝔰

_{j}_{∞}admits the following description. Consider the (commutative associative) algebra of truncated polynomials ℝ

_{4}[

*s*] = ℝ[

*s*]/(

*s*)

^{4}and the Lie algebra 𝔥 of Hamiltonian vector fields on ℝ

^{2}[15] Then 𝔰

_{∞}is isomorphic to the Lie algebra ℝ

_{4}[

*s*] ⊗ 𝔥 with the bracket [

*f*⊗

*V,g*⊗

*W*] =

*f g*⊗ [

*V,W*] for

*f ,g*∈ ℝ

_{4}[

*s*] and V, W ∈ 𝔥.

## 4. Infinite-dimensional covering, shadows, and nonlocal symmetries

Substituting

Direct computations prove:

### Proposition 4.1.

*Function* υ = *q is a shadow in the covering (1.2)*.

Then we have:

### Corollary 4.1.

*Functions* υ* _{k}* =

*q*,

_{k}*k*≥ 0,

*are shadows in the covering (4.1).*

A nonlocal symmetry of Eq. (1.1) is an infinite sequence (*φ*, *Q*_{0}, *Q*_{1},…, *Q _{m}*,…), where

*φ*=

*φ*(

*x*) and

^{i},u,u_{xi},...,q_{j},q_{j,x},q_{j,y},...*Q*=

_{m}*Q*(

_{m}*x*),

^{i},u,u_{xi},...,q_{j},q_{j,x},q_{j,y},...*m*≥ 0, are solutions to the equation

### Remark 4.1.

To simplify notation, here and below we use *φ _{xz}* for

The nonlocal symmetries of Eq. (1.1) are described by the following theorems:

### Theorem 4.1.

*The local symmetries φ*_{0}(*A*), …, *φ*_{3}(*A*), *ψ*_{1}, *ψ*_{2}, *ψ*_{3} have the lifts Φ_{0}(*A*), …, Φ_{3}(*A*), Ψ_{1}, Ψ_{2}, Ψ_{3} *to the nonlocal symmetries in the covering (4.1) defined as* Φ* _{i}*(

*A*) = (

*φ*(

_{i}*A*), Φ

_{i}_{,0}(

*A*), Φ

_{i}_{,1}(

*A*), …, Φ

_{i}_{,}

*(*

_{k}*A*), …), Ψ

*= (*

_{j}*ψ*, Ψ

_{j}

_{j}_{,0}, Ψ

_{j}_{,1}, …, Ψ

_{j}_{,}

*, …), with*

_{k}The proof is similar to the proof of theorem 4.2 below and is therefore omitted.

### Theorem 4.2.

*The shadows v _{k}* =

*q*,

_{k}*k*≥ 0,

*have the lifts*ϒ

_{k}t

*o the nonlocal symmetries in the covering (4.1) defined as*ϒ

*= (*

_{k}*q*, ϒ

_{k}

_{k}_{,0}, ϒ

_{k}_{,1}, … , ϒ

_{k}_{,}

*, …)*

_{m}*with*

*where*⟨

*a*,

*b*⟩ =

*a*−

_{x}b_{y}*a*

_{y}b_{x}### Remark 4.2.

The bracket ⟨⋅, ⋅⟩ is a Lie bracket. It endows the space *C*^{∞}(ℝ^{2}) of smooth functions on ℝ^{2} with the structure of a Lie algebra with the noncentral part isomorphic to the algebra of Hamiltonian vector fields on ℝ^{2} and the center generated by the constant functions.

**Proof.** We claim that^{d}

*= ⟨ϒ*

_{k,m,z}*,*

_{k,m}*u*⟩+ ⟨

_{y}*q*,

_{m}*q*⟩+ϒ

_{k,y}

_{k, m}_{−1,}

*is analogus.*

_{y}## 5. The structure of the algebra of nonlocal symmetries

The structure of the algebra

### Theorem 5.1.

*The Jacobi brackets of lifts of the local symmetries are lifts of the Jacobi brackets of the corresponding local symmetries, that is, the commutators for* Φ* _{m}*(

*A*), Ψ

*(3.1)—(3.5)*

_{k}satisfy equations*with φ*(

_{m}*A*),

*ψ*Φ

_{k}replaced by*(*

_{m}*A*), Ψ

*,*

_{k}*respectively*.

*The other Jacobi brackets are*

**Proof.** We will prove that {Φ_{0}(*A*), ϒ* _{k}*} = 0 . The proof for the other Jacobi brackets is similar and therefore is omitted.

We start with writing down explicitly the formula for computing the Jacobi bracket in case of some infinite-dimensional vectors Θ and Ω. In order to facilitate applying this formula to vectors Φ_{0}(*A*) and ϒ* _{k}* we enumerate coordinates of the vectors Θ and Ω in the following way: Θ = (Θ

_{−1}, Θ

_{0},…, Θ

*,…), Ω = (Ω*

_{m}_{−1}, Ω

_{0}, …, Ω

*,…) . Then the Jacobi bracket is of the form {Θ, Ω} = ({Θ, Ω}*

_{m}_{−1},{Θ, Ω}

_{0},{Θ, Ω}

_{1},…), where

*q*

_{m}_{,}

*, Φ*

_{I}_{0,}

*(*

_{j}*A*) depends at most on

*q*

_{j}_{,}

*,*

_{t}*q*

_{j}_{,}

*,*

_{x}*q*

_{j}_{,}

*,*

_{y}*q*

_{j}_{,}

*,*

_{z}*q*or

_{j}*q*

_{j}_{+ 1}and ϒ

_{k}_{,}

*depends at most on*

_{j}*q*

_{m}_{,}

*,*

_{x}*q*

_{k}_{+}

_{j}_{+ 1−}

_{m}_{,}

*,*

_{x}*q*

_{m}_{,}

*,*

_{y}*q*

_{k}_{+}

_{j}_{+ 1−}

_{m}_{,}

*for*

_{y}*m*= 0, …,

*j*. The following observations will be used frequently:

*i*= 0,1,2,3, hence {Φ

*(*

_{i}*A*), ϒ

*}*

_{k}_{−1}= 0 . Then for

*j*≥ 0 we get

### Remark 5.1.

Denote by 𝔞 the ideal of the Lie algebra * _{i}* and by 𝔟 the subalgebra generated by Γ

*. Identify 𝔰*

_{j}_{∞}and

*(*

_{m}*A*) and the subalgebra generated by Ψ

*, respectively. Then we obtain*

_{k}## 6. Conclusion

In this paper we have found two infinite hierarchies of commuting nonlocal symmetries for Plebański’s second heavenly equation. One of these hierarchies can be employed to construct new solutions using the techniques presented e.g. in [10,19,31], see also [5,16,17,34]. Furthermore, we find the lifts of local symmetries and the structure of the Lie algebra of the nonlocal symmetries. We emphasize that finding an explicit form of nonlocal symmetries and the commutator relations for an infinite-dimensional symmetry algebra of nonlocal symmetries (rather than just shadows) is quite rare. We know just a number of similar results in the literature, [1–3,5,13,16–18,29]. We hope that it would be very interesting to study how the infinite-dimensional Lie algebra of nonlocal symmetries reflects the algebraic structure behind the integrability properties of the considered equation, cf. [27,28].

## Acknowledgments

The authors gratefully acknowledge financial support from the Polish Ministry of Science and Higher Education. We are pleased to thank A. Sergyeyev for important and stimulating discussions, to E.G. Reyes for pointing out the references [5,16,17], and to the anonymous referee for useful suggestions.

## Footnotes

We identify here the Lie algebra of local symmetries with its lift to nonlocal symmetries in the covering (4.1)

see remark 4.1.

## References

### Cite this article

TY - JOUR AU - Aleksandra Lelito AU - Oleg I. Morozov PY - 2021 DA - 2021/01/06 TI - Nonlocal symmetries of Plebański’s second heavenly equation JO - Journal of Nonlinear Mathematical Physics SP - 188 EP - 197 VL - 25 IS - 2 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2018.1452669 DO - 10.1080/14029251.2018.1452669 ID - Lelito2021 ER -