On Stability of Standing Waves of Nonlinear Dirac Equations

Boussaïd, Nabile; Cuccagna, Scipio

doi:10.1080/03605302.2012.665973

Cited by 55 publications

(77 citation statements)

References 56 publications

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“…Generalizing the results on orbital stability of solitary waves [14] to the nonlinear Dirac equation does not seem realistic, because of the corresponding energy functional being sign-indefinite; instead, one hopes to prove the asymptotic stability, using linear stability combined with the dispersive estimates. The first results on asymptotic stability for the nonlinear Dirac equation are already appearing [1,16], with the assumptions on the spectrum of the linearized equation playing a crucial role. In view of these applications, the spectrum of the linearization at a solitary wave is of great interest.…”

Section: Introductionmentioning

confidence: 99%

On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

Berkolaiko

Comech

2012

Math. Model. Nat. Phenom.

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Abstract.We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.

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Section: Introductionmentioning

confidence: 99%

On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

Berkolaiko

Comech

2012

Math. Model. Nat. Phenom.

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“…Importantly, the equation has seen a significant volume of studies from a more mathematical perspective. Various aspects have been examined in this context, including the spectral stability and the potential emergence of point spectrum eigenvalues with nonzero real part (which has been shown to be impossible to happen beyond the so-called embedded thresholds) [11], the orbital and asymptotic stability under a series of relevant assumptions [12], the nonlinear Schrödinger (non-relativistic) limit and its instability for nonlinearities beyond a critical exponent [13], as well as classical (Vakhitov-Kolokolov) and more suitable to this setting (energy based) criteria [14] for the linear stability of solitary waves in the NLDE. A series of more computationally/physically oriented studies both in the context of the stability/dynamics of the NLDE solitary waves [15,16] (again, in principle for arbitrary nonlinearity powers) and in that of these structures in the presence of external fields [17] have also recently appeared.…”

Section: Introductionmentioning

confidence: 99%

Solitary waves in a discrete nonlinear Dirac equation

Cuevas–Maraver¹,

Saxena²

2015

J. Phys. A: Math. Theor.

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In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross-Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two-and three-site excitations playing a role analogous to one-and twosite, respectively, excitations of the discrete nonlinear Schrödinger analogue of the model. Stability exchanges between the two-and three-site states are identified, as well as instabilities that appear to be persistent over the coupling strength , for a subcritical value of the propagation constant Λ. Variations of the propagation constant, coupling parameter and nonlinearity exponent are all examined in terms of their existence and stability implications and long dynamical simulations are used to unravel the evolutionary phenomenology of the system (when unstable).

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“…The question of stability of solitary waves is of utmost importance: perturbations ensure that we only ever encounter stable configurations. Recent attempts at asymptotic stability of solitary waves in the nonlinear Dirac equation [Bou06,Bou08,PS12,BC12c,Cuc12] rely on the fundamental question of spectral stability:…”

Section: Introductionmentioning

confidence: 99%

On linear instability of solitary waves for the nonlinear Dirac equation

Comech

Guan

Gustafson

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. We consider the nonlinear Dirac equation, also known as the Soler model:We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit ω → m, proving that if k > 2/n, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with ω sufficiently close to m, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh-Schrödinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion.Résumé. Nous considérons l'équation de Dirac non linéaire, aussi connu comme le modèle de Soler. Nous étudions le spectre ponctuel des linéarisations aux ondes solitaires des petites amplitudes dans la limite ω → m, et montrons que si k > 2/n, ensuite une valeur propre positive et une négative sont présents dans le spectre des linéarisations à ces ondes solitaires lorsque ω est suffisamment proche de m, ensuite ces ondes solitaires sont linéairement instable. L'approche est basée sur l'application de théorie de la perturbation de Rayleigh-Schrödinger à la limite non relativiste de l'équation. Les résultats sont en accord formel avec le critère de stabilité de Vakhitov-Kolokolov.

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On Stability of Standing Waves of Nonlinear Dirac Equations

Cited by 55 publications

References 56 publications

On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

Solitary waves in a discrete nonlinear Dirac equation

On linear instability of solitary waves for the nonlinear Dirac equation

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