Nonlocal symmetries related to B\"acklund transformation and their applications

TL;DR

This paper explores nonlocal symmetries derived from Bäcklund transformations for the potential KdV equation, leading to new hierarchies, solutions, and integrable models in finite and infinite dimensions.

Contribution

It introduces a new compact form of nonlocal symmetry from BT and applies it to construct hierarchies, solutions, and integrable models, expanding the understanding of symmetry methods.

Findings

Constructed negative pKdV hierarchies and bilinear forms

Localized nonlocal symmetry to generate new solutions

Derived finite-dimensional integrable models

Abstract

Starting from nonlocal symmetries related to B\"acklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact form which comes from BT is presented and used to make researches in the following three subjects: two sets of negative pKdV hierarchies and their corresponding bilinear forms are constructed; the nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions; some other models both in finite dimensions and infinite dimensions are generated by comprising the original BT and evolution under new nonlocal symmetry. The finite-dimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.