Nonlocal symmetries related to Bäcklund transformation and their applications. (English) Zbl 1248.37069

This paper introduces nonlocal symmetries related to Bäcklund transformations (BT) taking the potential KdV equation as an illustrative example. A new class of nonlocal symmetries is derived and the prolongation of the new nonlocal symmetries is found. The finite symmetry transformation and similarity reductions are computed to give novel exact solutions of the KdV equation. The process leading to two exact solutions is proposed. The authors present a detailed description of the new nonlocal symmetry of the pKdV equation and extend the nonlocal symmetry to be equivalent to a Lie point symmetry of some auxiliary prolonged system. The authors give a finite-dimensional system that is equivalent in the Liouville sense. Various integrable systems are constructed and discussed by means of the symmetry constraint method. By applying nonlocal symmetry on the BT of the pKdV equation, a finite-dimensional integrable system is given. Moreover, the introduction of an internal parameter as a new argument helps build two sets of infinite-dimensional models.


37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35Q53 KdV equations (Korteweg-de Vries equations)
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