Backlund Transformations for Darboux Integrable Differential Systems: Examples and Applications

TL;DR

This paper develops a group-theoretical framework for constructing Bäcklund transformations for Darboux integrable systems, providing detailed examples, new applications, and clarifying the scope of existing transformations.

Contribution

It introduces a general group-theoretical method for constructing Bäcklund transformations and applies it to Darboux integrable equations, including new examples and non-existence results.

Findings

Constructed Bäcklund transformations using group methods.

Provided detailed examples demonstrating the theory.

Proved a non-existence theorem for certain Bäcklund transformations.

Abstract

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of B\"acklund transformations. We then showed how this construction can be applied to construct B\"acklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new applications which demonstrate the theory. In particular our final example demonstrates how our group theoretical approach produces all the B\"acklund transformations in arXiv:0707.4408. We also prove, using group methods, a non-existence theorem for B\"acklund transformatons which disagrees with part 2 of Theorem 1 in arXiv:0707.4408.