Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems

Baran, H.; Krasil’shchik, I.; Morozov, Oleg I.; Vojčák, P.

doi:10.1080/14029251.2014.975532

Cited by 29 publications

(17 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [1], two other equations, u y u xy − u x u yy = e y u xx and u yy = (u x + x)u xy − u y (u xx + 2), were obtained as symmetry reductions of the universal hierarchy and the 3D rdDym equations, respectively, were obtained. We plan study these equations by the methods similar to the used above in the forthcoming publications.…”

Section: Discussionmentioning

confidence: 99%

“…We recall here basic facts from the theory of nonlocal symmetries (see [8]) and previous results on Equation (1) and its reductions, see [1,2].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [1], we described all two-dimensional symmetry reductions of this equation. All these reductions are either linearizable or exactly solvable, except for the two ones:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

Holba¹,

Krasil’shchik²,

Morozov³

et al. 2021

JNMP

Self Cite

View full text Add to dashboard Cite

Abstract. We consider the 3D equation u yy = u tx + u y u xx − u x u xy and its 2D reductions: (1) u yy = (u y + y)u xx − u x u xy − 2 (which is equivalent to the Gibbons-Tsarev equation) and (2) u yy = (u y + 2x)u xx + (y − u x )u xy − u x . Using reduction of the known Lax pair for the 3D equation, we describe nonlocal symmetries of (1) and (2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.

show abstract

Section: Discussionmentioning

confidence: 99%

“…We recall here basic facts from the theory of nonlocal symmetries (see [8]) and previous results on Equation (1) and its reductions, see [1,2].…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

Holba¹,

Krasil’shchik²,

Morozov³

et al. 2021

JNMP

Self Cite

View full text Add to dashboard Cite

show abstract

“…2. The Maple package Jets [11], initially developed by Marvan (2003), then also by Baran (2010), has been used for computing Hamiltonian operators for particular differential equations [9,8,10,12]. However, it does not contain any specific feature for integrability structures (like an implementation of the Schouten bracket).…”

Section: Given Two Hamiltonian Operatorsmentioning

confidence: 99%

Computing with Hamiltonian operators

Vitolo

2019

Computer Physics Communications

View full text Add to dashboard Cite

Hamiltonian operators for partial differential equations are ubiquitous in mathematical models of theoretical and applied physics. In this paper the new Reduce package CDE for computations with Hamiltonian operators is presented. CDE can verify the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators, and it can compute the Lie derivative of a Hamiltonian operator with respect to a vector field. More generally, it can compute with (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of mathematical or theoretical physics.

show abstract

“…In [1] we gave a complete description of symmetry reductions for four three dimensional systems: the universal hierarchy equation, the 3D rdDym equation, the modified Veronese web equation, and Pavlov's equation. The result comprised more than 30 equations, but the majority of them were either exactly solvable or linearized by the generalized Legendre transformations.…”

Section: Introductionmentioning

confidence: 99%

Integrability properties of some equations obtained by symmetry reductions

Baran¹,

Krasil’shchik²,

Morozov³

et al. 2021

JNMP

Self Cite

View full text Add to dashboard Cite

In our recent paper [1], we gave a complete description of symmetry reduction of four Lax-integrable (i.e., possessing a zero-curvature representation with a non-removable parameter) 3-dimensional equations. Here we study the behavior of the integrability features of the initial equations under the reduction procedure. We show that the ZCRs are transformed to nonlinear differential coverings of the resulting 2D-systems similar to the one found for the Gibbons-Tsarev equation in [12]. Using these coverings we construct infinite series of (nonlocal) conservation laws and prove their nontriviality. We also show that the recursion operators are not preserved under reductions.

show abstract

Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems

Cited by 29 publications

References 8 publications

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

Computing with Hamiltonian operators

Integrability properties of some equations obtained by symmetry reductions

Contact Info

Product

Resources

About

Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems

Cited by 29 publications

References 8 publications

2D reductions of the equation u yy = u tx + u yu xx − u xu xy and their nonlocal symmetries

2D reductions of the equation u yy = u tx + u yu xx − u xu xy and their nonlocal symmetries

Computing with Hamiltonian operators

Integrability properties of some equations obtained by symmetry reductions

Contact Info

Product

Resources

About

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries