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

In this paper, we consider the strong instability of standing waves for a fourth-order nonlinear Schrodinger equation with the mixed dispersions i psi t - gamma Delta(2)psi + mu Delta psi + vertical bar psi vertical bar(p)psi = 0, (t, x) is an element of [0, T*) x 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 derivative(2)(lambda) S-omega(u(lambda))vertical bar(lambda=1) <= 0, the ground state standing wave e(i omega t)u is strongly unstable by blow-up, where u(lambda)(x) = lambda(N/2) u(lambda x) and S-omega 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 mu > 0. (C) 2020 Elsevier Ltd. All rights reserved.