NLS-type equations from quadratic pencil of Lax operators: negative flows

TL;DR

This paper introduces a new integrable NLS-type model derived from a quadratic pencil of Lax operators, analyzing its spectral problem, conserved quantities, and potential for modeling higher order NLS equations.

Contribution

It formulates a novel NLS-type integrable system with a quadratic Lax operator and explores its spectral properties and generalizations for symmetric spaces.

Findings

Developed a Lax representation with quadratic and rational spectral dependence

Analyzed the spectral problem and Riemann-Hilbert formulation for the model

Discussed potential applications to higher order NLS equations

Abstract

We formulate and study an integrable model of Nonlinear Schr\"odinger (NLS)-type through its Lax representation, where one of the Lax operators is quadratic and the other has a rational dependence on the spectral parameter. We discuss the associated spectral problem, the Riemann-Hilbert problem formulation, the conserved quantities, as well as a generalisation for symmetric spaces. In addition we explore the possibilities for modelling with higher order NLS (HNLS) integrable equations and in particular, the relevance of the proposed system.