Backlund Transformation for Integrable Hierarchies: example - mKdV Hierarchy

TL;DR

This paper explicitly constructs the mKdV hierarchy, decomposes it into graded sub-hierarchies, and extends Backlund transformations to all odd graded equations, providing universal solution verification.

Contribution

It introduces a comprehensive construction of the mKdV hierarchy and generalizes Backlund transformations across all odd graded equations within the same algebraic framework.

Findings

Decomposition of mKdV hierarchy into positive and negative sub-hierarchies

Extension of Backlund transformations to all odd graded equations

Verification of solutions satisfying universal Backlund transformations

Abstract

In this note we present explicitly the construction of the mKdV hierarchy and show that it decomposes into positive and negative graded sub-hierarchies. We extend the construction of the Backlund transformation for the sinh-Gordon model to all other positive and negative odd graded equations of motion generated by the same affine algebraic structure. Some simple examples of solutions are explicitly verified to satisfy, in a universal manner, the Backlund transformations for the first few odd (positive and negative) sub-hierarchies.