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Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions. (English) Zbl 1437.35624

Summary: In this paper, we consider the strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions \[i \psi_t - \gamma \Delta^2 \psi + \mu \Delta \psi + | \psi |^p \psi = 0, \quad (t, x) \in [ 0, T^\ast) \times \mathbb{R}^N ,\] where \(\gamma > 0\) and \(\mu < 0\). 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 \(\partial_\lambda^2 S_\omega (u^\lambda) |_{\lambda = 1} \leq 0\), the ground state standing wave \(e^{i \omega t} u\) is strongly unstable by blow-up, where \(u^\lambda (x) = \lambda^{\frac{ N}{ 2}} u (\lambda x )\) and \(S_\omega\) is the action. This result is a complement to the result of D. Bonheure et al. [Trans. Am. Math. Soc. 372, No. 3, 2167–2212 (2019; Zbl 1420.35343)], where the strong instability of standing waves has been studied in the case \(\mu > 0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
35A15 Variational methods applied to PDEs
35B44 Blow-up in context of PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35B35 Stability in context of PDEs

Citations:

Zbl 1420.35343
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References:

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