Non-commutative NLS-type hierarchies: dressing & solutions

TL;DR

This paper develops a framework for solving non-commutative matrix NLS hierarchies using Darboux-dressing methods, deriving solutions, recursion relations, and conserved quantities, and explores links to quantum models.

Contribution

It introduces a universal Darboux-dressing scheme for matrix NLS hierarchies, deriving solutions, Lax pairs, and non-commutative Riccati equations, extending the theory to quantum-like models.

Findings

Derived solutions for matrix NLS hierarchy using Darboux-dressing.

Established recursion relations for Lax pairs in the hierarchy.

Solved non-commutative Riccati equation and identified conserved quantities.

Abstract

We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general $n^{th}$ order matrix-differential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutitave Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.