Spectral problem for a two-component nonlinear Schr\"odinger equation in $2+1$ dimensions: Singular manifold method and Lie point symmetries

TL;DR

This paper introduces an integrable two-component nonlinear Schrödinger equation in 2+1 dimensions, deriving its Lax pair via the singular manifold method and analyzing its Lie point symmetries to find similarity reductions and spectral parameters.

Contribution

It presents a new integrable 2+1 dimensional two-component nonlinear Schrödinger equation, constructs its Lax pair, and explores its symmetry structure and reductions.

Findings

Derived a three-component Lax pair for the equation.

Identified an infinite-dimensional Lie algebra of symmetries.

Found that the spectral parameter arises from a symmetry.

Abstract

An integrable two-component nonlinear Schr\"odinger equation in $2+1$ dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional Lie algebra. The main non-trivial similarity reductions associated to these symmetries are identified. The spectral parameter of the reduced spectral problem appears as a consequence of one of the symmetries.