Construction of Type-II Backlund Transformation for the mKdV Hierarchy

TL;DR

This paper systematically constructs Type-II Backlund transformations for the entire mKdV hierarchy using an algebraic approach, providing explicit solutions and scattering results that are model-independent.

Contribution

It introduces a universal algebraic method to derive Backlund transformations for the mKdV hierarchy from the sinh-Gordon theory, extending known transformations to all hierarchy levels.

Findings

Explicit Backlund transformations for positive and negative grade evolutions.

Solutions for vacuum-vacuum and vacuum-one-soliton transitions.

Model-independent scattering delay calculations.

Abstract

From an algebraic construction of the mKdV hierarchy we observe that the space component of the Lax operator play a role of an universal algebraic object. This fact induces the universality of a gauge transformation that relates two field configurations of a given member of the hierarchy. Such gauge transformation generates the Backlund transformation (BT). In this paper we propose a systematic construction of Backlund Transformation for the entire mKdV hierarchy form the known Type-II BT of the sinh-Gordon theory. We explicitly construct the BT of the first few integrable models associated to positive and negative grade-time evolutions. Solutions of these transformations for several cases describing the transition from vacuum-vacuum and the vacuum to one-soliton solutions which determines the value for the auxiliary field and the the Backlund parameter respectively, independently of the model. The same follows for the scattering of two one-soliton solutions. The resultant delay is determined by a condition independent of the model considered.