Nonlocal symmetries of Lax integrable equations: a comparative study

الأصل · EN

We continue here the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation uty = uₓ uxy - uy uxx, (2) the 3D Pavlov equation uyy = utx + uy uxx - uₓ uxy; (3) the universal hierarchy equation uyy = uₜ uxy - uy utx, and (4) the modified Veronese web equation uty = uₜ uxy - uy utx. 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.