Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

TL;DR

This paper explores quadratic bundles linked to Hermitian symmetric spaces, focusing on a multi-component Kaup-Newell equation, and demonstrates how dressing methods can generate reflectionless potentials for integrable hierarchies.

Contribution

It introduces a dressing approach for quadratic bundles associated with symmetric spaces, enabling construction of reflectionless potentials and solutions for related integrable equations.

Findings

Derived reflectionless potentials with zero boundary conditions.

Constructed fast decaying solutions including solitons and rational solutions.

Applied dressing method to a multi-component Kaup-Newell equation.

Abstract

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.