Nonlocal symmetries for bilinear equations and their applications

TL;DR

This paper explores nonlocal symmetries of bilinear KP and BKP equations, deriving hierarchies and transformations that reveal new integrable properties and potential applications in soliton theory.

Contribution

It introduces a novel approach to generate bilinear NKP and NBKP hierarchies from nonlocal symmetries, unifying positive and negative hierarchies.

Findings

Derived bilinear NKP and NBKP hierarchies from nonlocal symmetries.

Established bilinear Bäcklund transformations for specific flows.

Showed that hierarchies can be obtained from the same nonlocal symmetries.

Abstract

In this paper, nonlocal symmetries for the bilinear KP and bilinear BKP equations are re-studied. Two arbitrary parameters are introduced in these nonlocal symmetries by considering gauge invariance of the bilinear KP and bilinear BKP equations under the transformation $f\longrightarrow fe^{ax+by+ct}$. By expanding these nonlocal symmetries in powers of each of two parameters, we have derived two types of bilinear NKP hierarchies and two types of bilinear NBKP hierarchies. An impressive observation is that bilinear positive and negative KP and BKP hierarchies may be derived from the same nonlocal symmetries for the KP and BKP equations. Besides, as two concrete examples, we have deived bilinear B\"acklund transformations for $t_{-2}$-flow of the NKP hierarchy and $t_{-1}$-flow of the NBKP hierarchy. All these results have made it clear that more nice integrable properties would be found for these obtained NKP hierarchies and NBKP hierarchies. Since KP and BKP hierarchies have played an essential role in soliton theory, we believe that the bilinear NKP and NBKP hierarchies will have their right place in this field.