Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations

Lin, Chen

doi:10.1007/bf02677396

Cited by 21 publications

(7 citation statements)

References 9 publications

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“…Remarks. Note that the linear stability results match precisely the orbital stability results in [9], except at the endpoints |c| = √ 2 2 . At this point, we have linear stability, according to theorem 3, but it is unclear whether the wave is orbitally stable or not.…”

Section: Stability For the Kgz Systemsupporting

confidence: 74%

“…In fact, there exists a two-parameter system of solitary-travelling waves of which the family displayed above is a particular case [9]. In the same paper, the author shows orbital stability for such waves, provided…”

Section: Examplesmentioning

confidence: 86%

“…We would like to point out that the powerful methods of GSS [16,17] reduce (in most cases) the problem of orbital stability to checking certain conditions on the linearized functionals. We mention in this regard the papers [6,9,27,28], which treat models of interest to us in this paper. We also note that establishing orbital instability for a given problem seems to be harder and requires more problem specific efforts [27].…”

Section: Introductionmentioning

confidence: 99%

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Linear stability analysis for travelling waves of second order in time PDE's

Stanislavova

Stefanov

2012

Nonlinearity

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We study traveling waves ϕc of second order in time PDE's utt + Lu + N (u) = 0. The linear stability analysis for these models is reduced to the question for stability of quadratic pencils in the form λ 2 Id + 2cλ∂x + Hc, where Hc = c 2 ∂xx + L + N ′ (ϕc).If Hc is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of ϕc. More precisely, we introduce an explicitly computable index ω * (Hc) ∈ (0, ∞], so that the wave ϕc is stable if and only if |c| ≥ ω * (Hc). The results are applicable both in the periodic case and in the whole line case.The method of proof involves a delicate analysis of a function G, associated with H, whose positive zeros are exactly the positive (unstable) eigenvalues of the pencil λ 2 Id + 2cλ∂x + H. We would like to emphasize that the function G is not the Evans function for the problem, but rather a new object that we define herein, which fits the situation rather well.As an application, we consider three classical models -the "good" Boussinesq equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq case and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves (and for all powers of p in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of [2] and [1], which apply to the case p = 2. For the beam equation, we provide an explicit formula (depending of the function ϕ ′ c L 2 ), which works for all p and for both the periodic and the whole line cases.Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves. Informally, we have found that in all the examples that we have looked at, our theory goes through, whenever the Grillakis-Shatah-Strauss theory applies. We believe that the results in this paper (or a variation thereof) will enable the linear stability analysis as well as asymptotic stability analysis for most models in the form utt + Lu + N (u) = 0.

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Section: Stability For the Kgz Systemsupporting

confidence: 74%

Section: Examplesmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

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Linear stability analysis for travelling waves of second order in time PDE's

Stanislavova

Stefanov

2012

Nonlinearity

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“…They describe the interaction of Langmuir waves and ion-acoustic waves in plasmas [8,9]. In order to seek soliton solutions to (1) and (2), the starting assumptions are [1][2][3][5][6][7][8][9] qðx; tÞ ¼ A 1 cosh…”

Section: (1+1) Dimensionsmentioning

confidence: 99%

“…The theory of nonlinear evolution equation (NLEEs) has made a profound progress in the last couple of decades [1][2][3][4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

1-Soliton solution of the Klein–Gordon–Zakharov equation with power law nonlinearity

Ismail

Biswas

2010

Applied Mathematics and Computation

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Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

Zheng

Shang

Peng

2016

Math Methods in App Sciences

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This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations utt−uxx+u+αuv+β|u|2u=0,vtt−vxx=(|u|2)xx, where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

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Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations

Cited by 21 publications

References 9 publications

Linear stability analysis for travelling waves of second order in time PDE's

Linear stability analysis for travelling waves of second order in time PDE's

1-Soliton solution of the Klein–Gordon–Zakharov equation with power law nonlinearity

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

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