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Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nyzhnyk equation

Oleksandra O. Vinnichenko, Vyacheslav M. Boyko, Roman O. Popovych

TL;DR

The paper addresses a distinguished submodel of the real symmetric potential dispersionless Nyzhnyk equation that is tied to a parametrized family of inequivalent one-dimensional subalgebras by an arbitrary time function. It shows this submodel is Lie-remarkable and that its point-symmetry pseudogroup is precisely the stabilizer of its maximal Lie invariance algebra, enabling a thorough symmetry-based analysis and reductions. By a two-step Lie-reduction framework, it constructs large families of exact invariant solutions linked to the inviscid Burgers equation, and provides an exhaustive description of generalized symmetries, cosymmetries, and conservation laws, including hidden nonlocal structures induced by the Burgers substitution. The work further develops the relation between Nyzhnyk models and Burgers-type reductions, illustrates the induction of symmetries along reductions, and extends the analysis to the intermediate equation, delivering a comprehensive view of symmetry-like objects across submodels with potential implications for integrable structures and exact solutions in higher dimensions.

Abstract

Among Lie submodels of the (real symmetric potential) dispersionless Nyzhnyk equation, we single out a remarkable submodel as such that, despite being the only one, is associated with a family of in general inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra of this equation, which are parameterized by an arbitrary function of the time variable. The large family of invariant solutions of the dispersionless Nyzhnyk equation that are related to the above submodel is expressed in terms of an arbitrary function of the time variable and the double quadrature of the well-known (implicit) general solution of the inviscid Burgers equation with respect to a space-like submodel invariant variable. The singled out submodel possesses many other interesting properties. In particular, we show that it is Lie-remarkable, and its maximal Lie invariance algebra completely defines its point symmetry pseudogroup, which provides the second but simpler example of the latter phenomenon in literature. Moreover, only hidden Lie symmetries of the dispersionless Nyzhnyk equation that are associated with this submodel are essential for finding its exact solutions. Using Lie reductions, we construct new families of exact solutions of the inviscid Burgers equation and the dispersionless Nyzhnyk equation in closed or parametric form. We also exhaustively described generalized symmetries, cosymmetries and conservation laws of the submodel, which gives the corresponding nonlocal and hidden structures for the inviscid Burgers equation and the dispersionless Nyzhnyk equation, respectively.

Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nyzhnyk equation

TL;DR

The paper addresses a distinguished submodel of the real symmetric potential dispersionless Nyzhnyk equation that is tied to a parametrized family of inequivalent one-dimensional subalgebras by an arbitrary time function. It shows this submodel is Lie-remarkable and that its point-symmetry pseudogroup is precisely the stabilizer of its maximal Lie invariance algebra, enabling a thorough symmetry-based analysis and reductions. By a two-step Lie-reduction framework, it constructs large families of exact invariant solutions linked to the inviscid Burgers equation, and provides an exhaustive description of generalized symmetries, cosymmetries, and conservation laws, including hidden nonlocal structures induced by the Burgers substitution. The work further develops the relation between Nyzhnyk models and Burgers-type reductions, illustrates the induction of symmetries along reductions, and extends the analysis to the intermediate equation, delivering a comprehensive view of symmetry-like objects across submodels with potential implications for integrable structures and exact solutions in higher dimensions.

Abstract

Among Lie submodels of the (real symmetric potential) dispersionless Nyzhnyk equation, we single out a remarkable submodel as such that, despite being the only one, is associated with a family of in general inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra of this equation, which are parameterized by an arbitrary function of the time variable. The large family of invariant solutions of the dispersionless Nyzhnyk equation that are related to the above submodel is expressed in terms of an arbitrary function of the time variable and the double quadrature of the well-known (implicit) general solution of the inviscid Burgers equation with respect to a space-like submodel invariant variable. The singled out submodel possesses many other interesting properties. In particular, we show that it is Lie-remarkable, and its maximal Lie invariance algebra completely defines its point symmetry pseudogroup, which provides the second but simpler example of the latter phenomenon in literature. Moreover, only hidden Lie symmetries of the dispersionless Nyzhnyk equation that are associated with this submodel are essential for finding its exact solutions. Using Lie reductions, we construct new families of exact solutions of the inviscid Burgers equation and the dispersionless Nyzhnyk equation in closed or parametric form. We also exhaustively described generalized symmetries, cosymmetries and conservation laws of the submodel, which gives the corresponding nonlocal and hidden structures for the inviscid Burgers equation and the dispersionless Nyzhnyk equation, respectively.
Paper Structure (16 sections, 24 theorems, 179 equations, 1 table)

This paper contains 16 sections, 24 theorems, 179 equations, 1 table.

Table of Contents

  1. Introduction
  2. Maximal Lie invariance algebra
  3. Point symmetry pseudogroup
  4. Defining properties of Lie symmetries
  5. On induction of Lie and point symmetries
  6. Classification of appropriate one-dimensional subalgebras
  7. Lie invariant solutions
  8. Local symmetry-like objects
  9. Integrals
  10. Auxiliary results
  11. Generalized symmetries
  12. Cosymmetries
  13. Conservation laws
  14. Relation to symmetry-like objects of inviscid Burgers equation
  15. Symmetry-like objects of the intermediate equation
  16. ...and 1 more sections

Key Result

Lemma 1

The radical $\mathfrak r$ of $\mathfrak a_{1.3}$ coincides with $\langle D^2,\,P^2,\,H,\,R(\alpha),\,Z(\sigma)\rangle$.

Theorems & Definitions (52)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Corollary 7
  • proof
  • ...and 42 more