Global search for localised modes in scalar and vector nonlinear Schr\"odinger-type equations

TL;DR

This paper introduces a novel numerical approach to find localized modes in scalar and vector nonlinear Schrödinger equations by analyzing singularities of solutions and classifying bounded solutions based on initial data.

Contribution

The authors develop a new method that identifies localized solutions by examining singularities and asymptotic behavior, applicable to various Schrödinger-type equations.

Findings

Successfully identified localized modes in multiple equations

Provided a numerical framework for classifying bounded solutions

Demonstrated the method on physical models like Gross-Pitaevskii and Lugiato-Lefever equations

Abstract

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schr\"odinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.