Non-commutative NLS-type hierarchies: dressing & solutions

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.

arXiv comment29 pages, LaTex. Minor modifications

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• Grassmannian flows and applications to non-commutative non-local and local integrable systems
• Time-like boundary conditions in the NLS model
• A Dual Construction of the Isotropic Landau-Lifshitz Model
• An algebraic approach to discrete time integrability
• Discretizations of the generalized AKNS scheme
• Time evolution in quantum systems and stochastics