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arXiv · 1810.10937

Non-commutative NLS-type hierarchies: dressing & solutions

Anastasia Doikou·Iain Findlay·Spyridoula Sklaveniti
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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.

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BibTeXRIS

Anastasia Doikou, Iain Findlay, Spyridoula Sklaveniti. 2018-10-25. Non-commutative NLS-type hierarchies: dressing & solutions. https://doi.org/10.1016/j.nuclphysb.2019.02.019

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