Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

In this paper, we consider the strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions iψt−γΔ2ψ+μΔψ+|ψ|pψ=0,(t,x)∈[0,T∗)×RN,where γ>0 and μ0.

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Published in:Nonlinear analysis 2020-07, Vol.196, p.111791, Article 111791
Main Authors: Feng, Binhua, Liu, Jiayin, Niu, Huiling, Zhang, Binlin
Format: Article
Language:English
Subjects:
Bi-harmonic nonlinear Schrödinger equation
Dispersions
Ground state
Laser beams
Nonlinearity
Schrodinger equation
Stability
Standing waves
Strong instability
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Liu, Jiayin
Niu, Huiling
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description In this paper, we consider the strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions iψt−γΔ2ψ+μΔψ+|ψ|pψ=0,(t,x)∈[0,T∗)×RN,where γ>0 and μ0.
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This equation arises in describing the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. We firstly obtain the variational characterization of ground state solutions by using the profile decomposition theory in H2. Then, we deduce that if ∂λ2Sω(uλ)|λ=1≤0, the ground state standing wave eiωtu is strongly unstable by blow-up, where uλ(x)=λN2u(λx) and Sω is the action. This result is a complement to the result of Bonheure et al. 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subjects Bi-harmonic nonlinear Schrödinger equation
Dispersions
Ground state
Laser beams
Nonlinearity
Schrodinger equation
Stability
Standing waves
Strong instability
title Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions
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