Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case

Abstract

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in [16] to show the orbital instability of the solitary waves Qω,c (x − ct) e i ωt of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case c = 2z0√ ω, where z0 = z0 (σ) is the unique zero point of F (z; σ) in (−1, 1). The new ingredients in the proof are the refined modulation decomposition of the solution near Qω,c according to the spectrum property of the linearized operator S ω,c (Qω,c) and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in [7].

Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case

“…Proof. The proof is similar to that of [20,Proposition 4.1]. We give the details for the reader's convenience.…”

“…Fukuizumi, Ohta and Ozawa conjectured that the solitary wave e i ωt Q ω (x) with ω = Ω(p, γ) is orbitally unstable in [8]. The purpose of this paper is to prove this conjecture according to the observations in [5,15,16,20,21]. More precisely, we have the main result as following.…”

“…It corresponds to the degenerate case, and the Vakhitov-Kolokolov stability criterion in [27] (see also [2,10,26]) breaks down in this case. In fact, one can still consider the stability (or instability) of the solitary waves through the non-degenerate behavior of higher order derivative of d (ω) as those in [16] (see also [5,15,20,21]). For this purpose, we first characterize the behavior of d (ω) at the critical value Ω.…”

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In this paper, we show the orbital instability of the solitary waves QΩe iΩt of the 1d NLS with an attractive delta potential (γ > 0)where Ω = Ω(p, γ) > γ 2 4 is the critical oscillation number and determined byThe classical convex method and Grillakis-Shatah-Strauss's stability approach in [2,10] don't work in this degenerate case, and the argument here is motivated by those in [5,15,16,20,21].The main ingredients are to construct the unstable second order approximation near the solitary wave QΩe iΩt on the level set M(QΩ) accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves QΩe iΩt in the energy space. Our result is the complement of the results in [8] in the degenerate case.

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“…The classical convex method and Grillakis-Shatah-Strauss's stability approach in [2,10] don't work in this degenerate case, and the argument here is motivated by those in [5,15,16,20,21].…”