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Article Dans Une Revue Complex Variables and Elliptic Equations Année : 2022

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

Résumé

We consider the following triple power nonlinear Schrödinger equation: iut + ∆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√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 > −ε for ε > 0 small enough in the case n = 3, we prove that standing waves with positive frequency are unstable by blow up.
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Dates et versions

hal-03379146 , version 1 (14-10-2021)
hal-03379146 , version 2 (21-03-2022)

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Phan van Tin. Instability of algebraic standing waves for nonlinear Schr\"odinger equations with triple power nonlinearities. Complex Variables and Elliptic Equations, 2022, ⟨10.1080/17476933.2022.2146104⟩. ⟨hal-03379146v2⟩
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