Stability theory for solitary-wave solutions of scalar field equations

Henry, Daniel; Perez, J. Fernando; Wreszinski, Walter F.

doi:10.1007/bf01208719

Cited by 102 publications

(112 citation statements)

References 11 publications

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“…(2) We can adapt the results due to Henry-Perez-Wreszinski [19] to the case of dispersive equations and to obtain that the linear instability result above implies the nonlinear instability. The implication linear to nonlinear is obtained because the mapping data-solution associated with the KdV system (1.30) is at least of class 2 (see Colliander-Keel-Sta lani-Takaoka-Tao [12]).…”

Section: (Non)linear Instability Resultsmentioning

confidence: 91%

(Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations

Pava¹,

Natali

2014

Advances in Nonlinear Analysis

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We prove the existence and nonlinear instability of periodic traveling wave solutions for the critical one-dimensional Klein-Gordon equation. We also establish a linear instability criterium for a KdV type system. An application of this approach is made to obtain the linear/nonlinear instability of vector cnoidal wave pro les. Finally, via a theoretical and numerical approach we show the linear stability or instability of periodic positive and sign changing waves, respectively, for the critical Korteweg-de Vries equation.

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Section: (Non)linear Instability Resultsmentioning

confidence: 91%

(Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations

Pava¹,

Natali

2014

Advances in Nonlinear Analysis

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“…Like NLS, the φ 4 equation is time reversible, as it is invariant under the transformation t → −t. It is also well posed for (φ(·, 0), φ t (·, 0)) ∈ H 2 loc × H 1 , decaying sufficiently rapidly to ±1 as |x| → ∞, see [30]. Hence, the reversal operator Q −1 (t) exists and is well-posed for all t > 0.…”

Section: Third Example-kink-antikink Collisions In the φ 4 Equationmentioning

confidence: 91%

Loss of physical reversibility in reversible systems

Sagiv

Ditkowski

Goodman

et al. 2020

Physica D: Nonlinear Phenomena

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A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schrödinger equation and the φ 4 equation can become "physically irreversible". By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural "arrow of time", reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.Hence, if y(t f ) = 0 at some time t f > 0, then y(0) cannot be uniquely determined.

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“…Arnold [5,6] in his study of incompressible ideal fluid flows. Since its introduction, the Lyapunov method has formed the crux of subsequent nonlinear stability techniques (see [29,30,52] for instance). We build on the results recently obtained for the cnoidal wave (genus one) solutions in [16], and present a systematic generalization.…”

Section: Introductionmentioning

confidence: 99%

Periodic finite-genus solutions of the KdV equation are orbitally stable

Nivala

Deconinck

2010

Physica D: Nonlinear Phenomena

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The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. The KdV equation is known to have large families of periodic solutions that are parameterized by hyperelliptic Riemann surfaces. They are generalizations of the famous multi-soliton solutions. We show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations: perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution.

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Stability theory for solitary-wave solutions of scalar field equations

Abstract: We prove stability and instability theorems for solitary-wave solutions of classical scalar field equations. * Partial financial support by FAPESP and FINEP. ** Partial financial support by CNPq.

Cited by 102 publications

References 11 publications

(Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations

(Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations

Loss of physical reversibility in reversible systems

Periodic finite-genus solutions of the KdV equation are orbitally stable

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