Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential

2018/04/06 by Noriyoshi Fukaya, Fukaya, Noriyoshi, Masahito Ohta +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.1804.02127

openalex publication_date 2018/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the strong instability of standing waves e^iωtϕ_ω(x) for nonlinear Schrödinger equations with an L²-supercritical nonlinearity and an attractive inverse power potential, where ω∈ℝ is a frequency, and ϕ_ω∈ H¹(ℝ^N) is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if ∂_λ²S_ω(ϕ_ω^λ)|_λ=1≤0, then the standing wave is strongly unstable, where S_ω is the action, and ϕ_ω^λ(x)\mathrel\mathop:=λ^N/2ϕ_ω(λx) is the scaling, which does not change the L²-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.

Citations

• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• A relation between pointwise convergence of functions and convergence of functionals

Cited by

• Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension

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