Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

TL;DR

This paper develops a multi-component modified KdV hierarchy using the Drinfel'd-Sokolov scheme, deriving its conservation laws, recursion operator, and exact solutions including solitons and breathers.

Contribution

It introduces a novel multi-component hierarchy with a non-regular Lax operator and provides explicit exact solutions using the rational dressing method.

Findings

Constructed the hierarchy and conservation laws based on Drinfel'd-Sokolov scheme.

Derived the recursion operator from the hierarchy's symmetry structure.

Obtained explicit one-soliton and breather solutions in determinant form.

Abstract

We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov scheme, however, in our case the Lax operator contains a constant non-regular element of the underlying Lie algebra. We also derive the associated recursion operator of the hierarchy using the symmetry structure of the Lax operators. Finally, using the rational dressing method, we obtain the one soliton solution, and we find the one breather solution of general rank in terms of determinants.