Instability of algebraic standing waves for nonlinear Schr\''odinger equations with triple power nonlinearities

TL;DR

This paper investigates the stability of algebraic standing waves in a nonlinear Schrödinger equation with triple power nonlinearities, revealing their instability under various focusing and defocusing conditions across different dimensions.

Contribution

It provides new results on the orbital instability and blow-up behavior of algebraic standing waves in the triple power nonlinear Schrödinger equation for specific parameter regimes.

Findings

Standing waves are orbitally unstable in certain focusing/defocusing cases for n=2,3.

Standing waves with positive frequency are unstable by blow-up in supercritical regimes.

Instability depends on the interplay of nonlinear powers and spatial dimensions.

Abstract

We consider the following triple power nonlinear Schr{\"o}dinger equation: iut + $\Delta$u + a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0. We are interested in algebraic standing waves i.e standing waves with algebraic decay above equation in dimensions n (n = 1, 2, 3). We prove the instability of these solutions in the cases DDF (we use abbreviation D: defocusing (ai < 0), F:focusing (ai > 0)) and DFF when n = 2, 3 and in the case DFF with a1 = --1, a3 = 1 and a2 < 32/15$\sqrt$6 when n = 1. Under these assumptions, the standing waves are orbitally unstable in the case of small positive frequency. When the highest power is L2(R^n)-supercritical power (for n = 2, 3), a1 = --1, a3 = 1 and a2 > --$\epsilon$ for $\epsilon$ > 0 small enough in the case n = 3, we prove that standing waves with positive frequency are unstable by blow up.