The vector form of Kundu-Eckhaus equation and its simplest solutions

TL;DR

This paper develops a hierarchy of integrable vector nonlinear equations, including a vector Kundu-Eckhaus equation, and presents new elliptic solutions and transformations between solutions.

Contribution

It introduces a new hierarchy of vector integrable equations depending on a functional parameter, extending known systems and providing explicit elliptic solutions.

Findings

Constructed a hierarchy of vector nonlinear equations using monodromy matrices.

Derived new elliptic solutions for the vector Kundu-Eckhaus and Manakov systems.

Showed existence of linear transformations between solutions of these vector equations.

Abstract

In our work a hierarchy of integrable vector nonlinear differential equations depending on the functional parameter $r$ is constructed using a monodromy matrix. The first equation of this hierarchy for $r=\alpha(\mathbf{p}^t\mathbf{q})$ is vector analogue of the Kundu-Eckhaus equation. When $\alpha=0$, the equations of this hierarchy turn into equations of the Manakov system hierarchy. New elliptic solutions to vector analogue of the Kundu-Eckhaus and Manakov system are presented. In conclusion, it is shown that there exist linear transformations of solutions to vector integrable nonlinear equations into other solutions to the same equations.