Energetic Stability Criterion for a Nonlinear Spinorial Model

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interactionand with mass m. Using the exact analytic form for rest frame solitary waves of the form (x,t) = ψ(x)e −iωt for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t c , it takes for the instability to set in, is an exponentially increasing function of ω and t c decreases monotonically with increasing κ.

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“…[16] are much too small to observe the instabilities we have found for ω < ω c . This also holds for the scattering experiments of Ref.…”

Section: Introductionmentioning

confidence: 71%

“…In this section, we present the numerical results for the Soler model [16], i.e., κ = 1. The first numerical simulation is performed using the OS(4) method for the two-hump wave with ω = 0.1; see Fig.…”

Section: A κ =mentioning

confidence: 99%

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

et al. 2014

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“…Let us notice that in the 3D case for the cubic nonlinearity f (s) = s (this is the original Soler model from [Sol70]), based on the numerical evidence from [Sol70,AS83], the charge Q(ω) has a local minimum at ω 1 ≈ 0.936m, suggesting that the solitary waves with ω 1 < ω < 1 are linearly unstable, but then at ω = ω 1 the real eigenvalues collide at λ = 0, and there are no nonzero real eigenvalues in the spectrum for ω ω 1 . Incidentally, this agrees with the "dilation-stability" results of [SV86] (one studies whether the energy is minimized or not under the chargepreserving dilation transformations).…”

Section: See Figurementioning

confidence: 99%

“…We mention the numerical simulations [AS83] and the analysis of the energy minimization under charge-preserving dilations and similar transformations [Bog79,AS86,SV86,CKMS10]. In spite of this, the question of spectral stability of solitary waves of nonlinear Dirac equation is still completely open.…”

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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“…In 1951, R. Finkelsten solved some rigorous solutions of the nonlinear Dirac equation by separating variables, and pointed out that the corresponding particles have quantized mass spectra [7,8]. The stability of the soliton has also been studied in [9][10][11], but opposite viewpoints have been proposed. The theoretical proof about the existence of solitons was provided in [12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Some properties of the spinor soliton

1998

AACA

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Abstraer. In this paper, the solitons of nonhnear Dirac equation ate discussed in detail, and several functions which reflect their characteristics ate computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 89and positive mass; the spinor soliton has two kinds of parity states , and each parity state has two ldnds of energy states; the larger the self-coupling coefficient w, the more the excitation states, and ii w is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.

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Energetic Stability Criterion for a Nonlinear Spinorial Model

Cited by 37 publications

References 11 publications

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

On linear instability of solitary waves for the nonlinear Dirac equation

Some properties of the spinor soliton

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