On the Multi-Component Nonlinear Schr\"odinger Equation with Constant Boundary Conditions

TL;DR

This paper develops an inverse scattering method for solving multi-component nonlinear Schrödinger equations on symmetric spaces with non-zero boundary conditions, introducing squared solutions as a nonlinear Fourier transform analogue.

Contribution

It generalizes the inverse scattering method using squared solutions for multi-component NLS equations on symmetric spaces, extending the Fourier transform analogy.

Findings

Derived the completeness relation for squared solutions

Provided spectral decomposition of recursion operators

Established the nonlinear Fourier transform framework

Abstract

The multi-component nonlinear Schrodinger equation related to C.I=Sp(2p)/U(p) and D.III=SO(2p)/U(p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use the generalized Zakharov-Shabat operator. We show that the ISM for L is a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual Fourier-exponential functions we use the so-called squared solutions, which are constructed in terms of the fundamental analytic solutions (FAS) of L and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. We derive the completeness relation for the squared solutions which turns out to provide spectral decomposition of the recursion (generating) operators, a natural generalizations of id/dx in the case of nonlinear evolution equations.