Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

TL;DR

This paper develops a Kac-Moody algebra framework for the Gardner equation within the mKdV hierarchy, enabling systematic construction of diverse soliton solutions including large amplitude and complex waveforms.

Contribution

It introduces a novel algebraic approach using deformed vertex operators to handle nonvanishing boundary conditions in the mKdV hierarchy.

Findings

Constructed explicit solutions including solitons, kinks, and breathers.

Extended the dressing method to nonvanishing boundary conditions.

Demonstrated the existence of large amplitude and complex wave solutions.

Abstract

A Kac-Moody algebra construction for the integrable hierarchy containing the Gardner equation is proposed. Solutions are systematically constructed employing the dressing method and deformed vertex operators which takes into account the nonvanishing boundary value problem for the mKdV hierarchy. Explicit examples are given and besides usual KdV like solitons, our solutions contemplate the large amplitude table-top solitons, kinks, dark solitons, breathers and wobbles.