Orbital stability of periodic traveling wave solutions for the Kawahara equation

Andrade, Thiago Pinguello de; Cristófani, Fabrício; Natali, Fábio

doi:10.1063/1.4980016

Cited by 10 publications

(17 citation statements)

References 14 publications

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“…(1.7) where p ≥ 1 is an integer. In fact, for the case p = 1, the authors in [3] established the orbital stability of explicit periodic traveling waves solution of the form…”

Section: Introductionmentioning

confidence: 99%

On the stability of periodic traveling waves for the modified Kawahara equation

Almeida

Cristófani²,

Natali

2019

Applicable Analysis

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In this paper, we present the first result concerning the orbital stability of periodic traveling waves for the modified Kawahara equation. Our method is based on the Fourier expansion of the periodic wave in order to know the behaviour of the nonpositive spectrum of the associated linearized operator around the periodic wave combined with a recent development which significantly simplifies the obtaining of orbital stability results.2000 Mathematics Subject Classification. 76B25, 35Q51, 35Q53.

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“…(1.7) where p ≥ 1 is an integer. In fact, for the case p = 1, the authors in [3] established the orbital stability of explicit periodic traveling waves solution of the form…”

Section: Introductionmentioning

confidence: 99%

On the stability of periodic traveling waves for the modified Kawahara equation

Almeida

Cristófani²,

Natali

2019

Applicable Analysis

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“…Namely, we wish to examine the spectral stability of waves arising as solutions of (1.4) and (1.5). Our constructions will not yield explicit waves 2 . Thus, we need to decide about their stability, based on their construction and properties.…”

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confidence: 91%

“…where d ≥ 1, p > 1, ε = ±1. These have been much studied, both in the NLS as well as Klein-Gordon context, since the early 90's, see for example [1,2]. For both models, we will be interested in the existence of solitons, and the corresponding close to soliton dynamics, in particular spectral stability.…”

mentioning

confidence: 99%

“…In fact, it depends on the parameter p, the sign of the parameter b, as well as the dimension d ≥ 1. Here, it is worth noting the works of Albert, [1] and Andrade-Cristofani-Natali, [2] in which the authors have mostly studied the stability of some explicitly available solutions in one spatial dimension. We proceed differently, by means of variational methods.…”

mentioning

confidence: 99%

“…They showed spectral stability for such small amplitude solutions. We should also mention the work [2], in which the authors consider the orbital stability for explicit periodic solutions of the Kawahara problem, subjected to a quadratic nonlinearity.…”

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confidence: 99%

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On the normalized ground states for the Kawahara equation and a fourth order NLS

Posukhovskyi¹,

Stefanov²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the Kawahara model and two fourth order semi-linear Schrödinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed thereinin all dimensions, for all acceptable values of the parameters. This, combined with the results of [5], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question. At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.

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Orbital stability of periodic traveling-wave solutions for a dispersive equation

Natali

2019

São Paulo J. Math. Sci.

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In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associated with the Kawahara equation.2000 Mathematics Subject Classification. 76B25, 35Q51, 35Q53.

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Orbital stability of periodic traveling wave solutions for the Kawahara equation

Cited by 10 publications

References 14 publications

On the stability of periodic traveling waves for the modified Kawahara equation

On the stability of periodic traveling waves for the modified Kawahara equation

On the normalized ground states for the Kawahara equation and a fourth order NLS

Orbital stability of periodic traveling-wave solutions for a dispersive equation

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