Uniqueness of standing-waves for a non-linear Schr\"odinger equation with three pure-power combinations in dimension one

TL;DR

This paper proves the uniqueness and non-degeneracy of symmetric ground-state standing waves for a one-dimensional non-linear Schrödinger equation with multi-power nonlinearities, extending previous results to broader classes.

Contribution

It establishes conditions under which ground-state solutions are unique and non-degenerate for a class of non-linearities combining multiple pure powers, including extensions beyond two or three powers.

Findings

Ground-state profiles are non-degenerate and unique up to translation and phase shift.

The class of nonlinearities satisfying these properties can be extended beyond simple power combinations.

Sufficient conditions involve an Euler differential inequality and properties of an auxiliary function.

Abstract

We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr\"odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.