Degeneracy and multiplicity of standing-waves of the one-dimensional non-linear Schr\"odinger equation for a class of algebraic non-linearities

TL;DR

This paper investigates the existence, stability, and degeneracy of standing-wave solutions in a one-dimensional nonlinear Schrödinger equation with algebraic non-linearities, revealing multiple minima and stability conditions.

Contribution

It introduces a class of algebraic nonlinearities and demonstrates the existence of degenerate minima, multiplicity of solutions, and stability properties of ground states.

Findings

Existence of degenerate minima for certain nonlinearities

Multiplicity of positive, radially symmetric minima with same mass and energy

Stability of ground-state and energy-minimizing standing-waves

Abstract

We study the existence, the stability and the non-degeneracy of normalized standing-waves solutions to a one dimensional non-linear Schr\"odinger equation. The non-linearity belongs to a class of algebraic functions appropriately defined. We can show that for some of these non-linearities one can observe the existence of degenerate minima, and the multiplicity of positive, radially symmetric minima having the same mass and the same energy. We also prove the stability of the ground-state and the stability of normalized standing-waves whose profile is a minimum of the energy constrained to the mass.