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On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierarchies. (English) Zbl 1284.35021

Summary: We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers’ hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35G20 Nonlinear higher-order PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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