Nonlocal symmetries of Lax integrable equations: a comparative study

Abstract

We continue here the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation $u_{ty}=u_x{u}_{xy}-u_y{u}_{xx}$, (2) the 3D Pavlov equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$; (3) the universal hierarchy equation $u_{yy}=u_t{u}_{xy}-u_y{u}_{tx}$, and (4) the modified Veronese web equation $u_{ty}=u_t{u}_{xy}-u_y{u}_{tx}$. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the for pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows (in the sense of [I.S.Krasil'shchik, A.M. Vinogradov, Acta Appl. Math., 15 (1989) 1-2, 161--209.]) of nonlocal symmetries.

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Cited by in corpus (3)

• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Reductions of the universal hierarchy and rdDym equations and their symmetry properties
• Recursion operators in the cotangent covering of the rdDym equation