Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

被引：0

作者：

H. Baran

I. S. Krasilshchik

O. I. Morozov

P. Vojčák

机构：

[1] Silesian University in Opava,Mathematical Institute

[2] Trapeznikov Institute of Control Sciences,Faculty of Applied Mathematics

[3] Independent University of Moscow,undefined

[4] AGH University of Science and Technology,undefined

来源：

Theoretical and Mathematical Physics | 2018年 / 196卷

关键词：

partial differential equation; integrable linearly degenerate equation; nonlocal symmetry; recursion operator;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation uty = uxuxy − uyuxx, (2) the Pavlov equation uyy = utx + uyuxx − uxuxy, (3) the universal hierarchy equation uyy = utuxy − uyutx, and (4) the modified Veronese web equation uty = utuxy − uyutx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.

引用

页码：1089 / 1110

页数：21

共 48 条

[1]

Baran H.(2014)Symmetry reductions and exact solutions of Lax integrable 3–dimensional systems J. Nonlinear Math. Phys. 21 643-671

[2]

Krasil’shchik I. S.(2015)Integrability properties of some equations obtained by symmetry reductions J. Nonlinear Math. Phys. 22 210-232

[3]

Morozov O. I.(1996)Reductions of the Benney equations Phys. Lett. A 211 19-24

[4]

Vojčák P.(2016)Coverings over Lax integrable equations and their nonlocal symmetries Theor. Math. Phys. 188 1273-1295

[5]

Baran H.(2015)Linearly degenerate partial differential equations and quadratic line complexes Commun. Anal. Geom. 23 91-127

[6]

Krasil’shchik I. S.(1989)Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations Acta Appl. Math. 15 161-209

[7]

Morozov O. I.(2002)Classical R–matrices on Poisson algebras and related dispersionless systems Phys. Lett. A 297 191-195

[8]

Vojčák P.(2010)Bi–Hamiltonian nature of the equation utx = uxyuy − uyyux Adv. Pure Appl. Math. 1 7-17

[9]

Gibbons J.(2006)The Kupershmidt hydrodynamics chains and lattices Internat. Math. Res. Not. 2006 46987-137

[10]

Tsarev S. P.(2004)A class of Einstein–Weil spaces associated to an integrable system of hydrodynamic type J. Geom. Phys. 51 126-4156

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