Nonlocal symmetries related to Bäcklund transformation and their applications

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 finitedimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.

Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

“…In the last few decades, many scientists and mathematicians have extensively studied the dynamic behaviors of several PDEs using different concepts. Symmetry analyses have been used to study the PDEs [2][3][4][5][6]. It has been found that some new solutions to PDEs can be obtained from the old ones through symmetry transformations [2].…”

New interaction solutions and nonlocal symmetries for the (2 + 1)-dimensional coupled Burgers equation

“…From the results of nonlocal symmetry reduction, Lou found that the symmetry related to the Painlev e truncated expansion is just the residue with respect to the singular manifold in the Painlev e analysis procedure and called residual symmetry (Gao, Lou, and Tang, 2013;Hu, Lou, and Chen, 2012;Lou, 2013;Lou, Hu, and Chen, 2012).…”

New nonlocal symmetries and conservation laws of the (1+1)-dimensional Sine-Gordon equation

“…For instance, nonlocal potential symmetries can be obtained through the local symmetries of the potential form of the given system, residual symmetries actually related to Bäcklund transformations (BTs) are obtained via the truncated Painlevé expansion [4]. Other types of well studied nonlocal symmetries include the so-called eigenfunction symmetries [5,6,7,8], and other nonlocal symmetries related to BTs [9].…”