A new extended matrix KP hierarchy and its solutions

TL;DR

This paper introduces a new extended matrix KP hierarchy with a novel $ au_B$ flow, deriving related integrable equations, and presents a generalized dressing method for constructing solutions, including solitons.

Contribution

It proposes a new extended matrix KP hierarchy with a $ au_B$ flow, expanding the class of integrable systems with self-consistent sources and solution methods.

Findings

Derived new 2+1 AKNS equations with self-consistent sources.

Obtained 2+1-dimensional Davey-Stewartson equations with sources.

Presented soliton solutions for these extended hierarchies.

Abstract

With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $\tau_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and ${\tau_B}$ and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its $t_A$-reduction and $\tau_B$ reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.