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), 643671; arXiv:1407.0246 [nlin.SI], DOI: 10.1080/14029251.2014.975532}], we gave a complete description of symmetry reduction of four Laxintegrable (i.e., possessing a zerocurvature representation with a nonremovable 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 2Dsystems similar to the one found for the GibbonsTsarev equation in [A.V. Odesskii, V.V. Sokolov, Nonhomogeneous 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.6461
 Bibcode:
 2014arXiv1412.6461B
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 35B06