Negative Even Grade mKdV Hierarchy and its Soliton Solutions

TL;DR

This paper develops an algebraic framework for the negative even mKdV hierarchy, introducing a modified dressing method and vertex operator to systematically construct explicit solutions for these equations.

Contribution

It provides a novel algebraic construction and a modified dressing method to generate explicit solutions for the negative even mKdV hierarchy.

Findings

Constructed a negative even mKdV hierarchy with algebraic methods.

Developed a modified dressing method for non-trivial vacuum configurations.

Obtained explicit solutions for the negative even grade equations.

Abstract

In this paper we provide an algebraic construction for the negative even mKdV hierarchy which gives rise to time evolutions associated to even graded Lie algebraic structure. We propose a modification of the dressing method, in order to incorporate a non-trivial vacuum configuration and construct a deformed vertex operator for $\hat{sl}(2)$, that enable us to obtain explicit and systematic solutions for the whole negative even grade equations.