Bound states spectrum of the nonlinear Schr\"odinger equation with P\"oschl-Teller and square potential wells

TL;DR

This paper analyzes the bound states spectrum of the nonlinear Schrödinger equation with P"oschl-Teller and square potential wells, revealing finite multi-node localized states and their properties through analytical and scattering methods.

Contribution

It introduces a detailed spectral analysis of bound states in nonlinear Schr"odinger equations with specific potentials, including modulational instability and scattering insights.

Findings

Finite number of multi-node localized states for fixed norm

Maximum number of nodes related to potential width

Existence of trapped modes confirmed by scattering experiments

Abstract

We obtain the spectrum of bound states for a modified P\"oschl-Teller and square potential wells in the nonlinear Schr\"odinger equation. For a fixed norm of bound states, the spectrum for both potentials turns out to consist of a finite number of multi-node localized states. We use modulational instability analysis to derive the relation that gives the number of possible localized states and the maximum number of nodes in terms of the width of the potential. Soliton scattering by these two potentials confirmed the existence of the localized states which form as trapped modes. Critical speed for quantum reflection was calculated using the energies of the trapped modes.