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Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation. (English) Zbl 1485.35349

Summary: In this paper, we consider the instability of standing waves for an inhomogeneous Gross-Pitaevskii equation \(i\psi_t +\Delta \psi -a^2|x|^2\psi +|x|^{-b}|\psi|^p\psi = 0.\) This equation arises in the description of nonlinear waves such as propagation of a laser beam in the optical fiber. We firstly proved that there exists \(\omega_* > 0\) such that for all \(\omega > \omega_*\), the standing wave \(\psi(t, x) = e^{i\omega t}u_\omega(x)\) is unstable. Then, we deduce that if \(\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0\), the ground state standing wave \(e^{i\omega t}u_\omega(x)\) is strongly unstable by blow-up, where \(u_\omega^\lambda(x) = \lambda^{\frac{N}{2}}u_\omega(\lambda x)\) and \(S_\omega\) is the action. This result is a complement to the partial result of A. H. Ardila and V. D. Dinh [Z. Angew. Math. Phys. 71, No. 3, Paper No. 79, 24 p. (2020; Zbl 1437.35615)], where the strong instability of standing waves has been studied under a different assumption.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35C08 Soliton solutions

Citations:

Zbl 1437.35615
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References:

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