Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schr{\"o}dinger equation

TL;DR

This paper provides a comprehensive classification of the stability and instability of standing waves in a double power nonlinear Schrödinger equation, using slope criteria and numerical validation.

Contribution

It introduces a reformulation of the slope criterion and explicitly calculates slope values at zero frequency, filling gaps in previous stability analyses.

Findings

Complete stability classification for positive frequency standing waves.

Explicit slope calculations at zero frequency.

Numerical experiments supporting theoretical results.

Abstract

For the double power one dimensional nonlinear Schr{\"o}dinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.