Nonlocal symmetries related to Bäcklund transformation and their applications
arXiv:1201.3409
Abstract
Starting from nonlocal symmetries related to Bäcklund 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.