Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions. (July 2020) - Feng, Binhua; Liu, Jiayin; Niu, Huiling; Zhang, Binlin
Abstract: In this paper, we consider the strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions i ψ t − γ Δ 2 ψ + μ Δ ψ + | ψ | p ψ = 0, ( t, x ) ∈ [ 0, T ∗ ) × R N, where γ > 0 and μ < 0 . 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 H 2 . Then, we deduce that if ∂ λ 2 S ω ( u λ ) | λ = 1 ≤ 0, the ground state standing wave e i ω t u is strongly unstable by blow-up, where u λ ( x ) = λ N 2 u ( λ x ) and S ω is the action. This result is a complement to the result of Bonheure et al. (2019), where the strong instability of standing waves has been studied in the case μ > 0 .