Nonlocal conservation laws and related B\"acklund transformations via reciprocal transformations

TL;DR

This paper explores the role of nonlocal conservation laws and reciprocal transformations in integrable systems, revealing new connections and auto-B"acklund transformations among (1+1)-dimensional evolution equations like KdV and Dym.

Contribution

It uncovers infinitely many nonlocal conservation laws and demonstrates how reciprocal transformations relate different integrable systems and their solutions.

Findings

Different nonlocal conservation laws can lead to the same integrable system via reciprocal transformation.

Reciprocal transformations can generate new solutions from existing ones.

Multiple conservation laws can produce auto-B"acklund transformations.

Abstract

A set of infinitely many nonlocal conservation laws are revealed for (1+1)-dimensional evolution equations. For some special known integrable systems, say, the KdV and Dym equations, it is found that different nonlocal conservation laws can lead to same new integrable systems via reciprocal transformation. On the other hand, it can be considered as one solution of the new model obtained via reciprocal transformation(s) can be changed to different solutions of the original model. The fact indicates also that two or more different (local and nonlocal) conservation laws can be used to find implicit auto-B\"acklund transformations via reciprocal transformation to other systems.