Orbital Instability of Standing Waves for the Klein-Gordon-Zakharov System

Abstract

This paper deals with the instability of the ground state solitary wave solution to the Klein-Gordon-Zakharov system in three space dimensions with c ≥ 1, which is a model to describe the Langmuir turbulence in plasma. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational calculus and scaling argument. Then by defining invariant sets and applying some priori estimates, we prove the orbital instability of the ground state in the following sense: in each neighborhood of it, there exists a solution whose energy diverges in finite or infinite time.

Long-time behaviour for a non-autonomous Klein-Gordon-Zakharov system

“…For a deeper and more detailed discussion about systems consisting of wave equations and other types of physical models, we refer to [16], [17], [22], [23] and [27].…”

Orbital stability for the periodic Zakharov system

“…The existence and orbital stability of solitary wave solutions for (1.1) (φ ω,c (ξ ), ψ ω,c (ξ ) → 0 as |ξ | → ∞) and some associated models have been studied from several points of view, see, for instance, Hui [34,35], Lin [41], Ohta [45], Wu [54], Zaihui [55] and Zaihui and Yi [56].…”

Instability of standing wave, global existence and blowup for the Klein–Gordon–Zakharov system with different-degree nonlinearities

“…When N = 3, there have been some works for the Klein-Gordon-Zakharov system (1.1)-(1.3) with the absence of the term −|u| 2 u (see [4,5,11,13,18,21]): c −2 u tt − u + c 2 u = −nu, t > 0, x ∈ R 3 , (1.5) α −2 n tt − n = |u| 2 , t > 0, x ∈ R 3 , (1.6) u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R 3 , n(0, x) = n 0 (x), n t (0, x) = n 1 (x), x ∈ R 3 .…”

“…Gan and Zhang [4,5] proved the existence of standing wave with ground state of (1.5)-(1.6) by applying an intricate variational argument and obtained the instability of the standing wave by applying Payne and Sattinger's potential well argument (see Payne and Sattinger [15]) and Levine's concavity method (see Levine [8]). Zhang and Gan [21] derived a sharp condition of global existence for the Cauchy problem (1.5)-(1.7).…”

“…Since the system (1.1)-(1.3) includes different-degree nonlinearities and derivative nonlinearity, in order to get the results in this paper aforementioned, we must improve some techniques different from those proposed in the papers studying the Klein-Gordon-Zakharov system with quadratic nonlinearity and the nonlinear coupled system of the Klein-Gordon and wave equations without derivative and different-degree nonlinearities (see [4,5,12,21,22]). On the one hand, for the derivative nonlinearity, we must introduce a homogeneous Sobolev spaceḢ −1 (R N ) which is defined bẏ…”