Integrability properties of some symmetry reductions

Abstract

In our recent paper [H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Voj{č}{á}k, Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems, Journal of Nonlinear Mathematical Physics, Vol. 21, No. 4 (December 2014), 643--671; arXiv:1407.0246 [nlin.SI], DOI: 10.1080/14029251.2014.975532}], 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 [A.V. Odesskii, V.V. Sokolov, Non-homogeneous systems of hydrodynamic type possessing Lax representations, arXiv:1206.5230, 2006]. 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.

References in corpus (5)

• Geometry of jet spaces and integrable systems
• Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems
• Recursion Operators and Nonlocal Symmetries for Integrable rmdKP and rdDym Equations
• Integrability of the Manakov--Santini hierarchy
• Systems of Gibbons-Tsarev type and integrable 3-dimensional models

Cited by in corpus (4)

• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Nonlocal symmetries of Lax integrable equations: a comparative study
• Reductions of the universal hierarchy and rdDym equations and their symmetry properties
• On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems