## Abstract

We consider the generalized matrix non-linear Schrödinger (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-commutative 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.

Original language | English |
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Pages (from-to) | 376-400 |

Number of pages | 25 |

Journal | Nuclear Physics B |

Volume | 941 |

Early online date | 26 Feb 2019 |

DOIs | |

Publication status | Published - Apr 2019 |

## Keywords

- math-ph
- hep-th
- math.MP
- nlin.SI

## ASJC Scopus subject areas

- Nuclear and High Energy Physics