Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy

Soler, Mario

doi:10.1103/physrevd.1.2766

Cited by 196 publications

(170 citation statements)

References 5 publications

Supporting

Mentioning

165

Contrasting

Unclassified

Order By: Relevance

“…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%

“…The existence of standing waves in the nonlinear Dirac equation was studied in [Sol70], [CV86], [Mer88], and [ES95]. The question of stability of solitary waves is of utmost importance: perturbations ensure that we only ever encounter stable configurations.…”

Section: Introductionmentioning

confidence: 99%

“…A natural simplification of the Dirac-Maxwell system [Gro66] is the nonlinear Dirac equation, such as the massive Thirring model [Thi58] with vector-vector self-interaction and the Soler model [Sol70] with scalar-scalar selfinteraction (known in dimension n = 1 as the massive Gross-Neveu model [GN74,LG75]). These models with self-interaction of local type have been receiving a lot of attention in particle physics (see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On linear instability of solitary waves for the nonlinear Dirac equation

Comech

Guan

Gustafson

2014

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

View full text Add to dashboard Cite

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.

show abstract

Section: See Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On linear instability of solitary waves for the nonlinear Dirac equation

Comech

Guan

Gustafson

2014

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

View full text Add to dashboard Cite

show abstract

“…Classical solutions of nonlinear field equations have a long history as a model of extended particles [12]. In 1970, Soler proposed that the self-interacting 4-Fermi theory was an interesting model for extended fermions.…”

Section: Introductionmentioning

confidence: 99%

Approximate analytic solutions to coupled nonlinear Dirac equations

2017

View full text Add to dashboard Cite

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalarscalar self interactions g 2 1 2 (ψψ) 2 + g 2 2 2 (φφ) 2 + g 2 3 (ψψ)(φφ) as well as vector-vector interactions of the form g 2 1 2 (ψγµψ)(ψγ µ ψ) + g 2 2 2 (φγµφ)(φγ µ φ) + g 2 3 (ψγµψ)(φγ µ φ). Writing the two components of the assumed solitary wave solution of these equation in the form ψ = e −iω 1 t {R1 cos θ, R1 sin θ}, φ = e −iω 2 t {R2 cos η, R2 sin η}, and assuming that θ(x), η(x) have the same functional form they had when g3=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 2 3 /g 2 2 and g 2 3 /g 2 1 . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.

show abstract

“…The ansatz (1.2) is well-known in physics and has been used in particular by Soler [35] to describe elementary fermions.…”

Section:   mentioning

confidence: 99%

A variational study of some hadron bag models

Treust

2013

Calc. Var.

View full text Add to dashboard Cite

Abstract. We study, in this paper, some relativistic hadron bag models. We prove the existence of excited state solutions in the symmetric case and of a ground state solution in the non-symmetric case for the soliton bag and the bag approximation models by concentration compactness. We show that the energy functionals of the bag approximation model are Γ-limits of sequences of soliton bag energy functionals for the ground and excited state problems. The pre-compactness, up to translation, of the sequence of ground state solutions associated with the soliton bag energy functionals in the non-symmetric case is obtained combining the Γ-convergence theory and the concentrationcompactness principle. Finally, we give a rigorous proof of the original derivation of the M.I.T. bag equations via a limit of bag approximation ground state solutions in the spherical case. The supersymmetry property of the Dirac operator is a key point in many of our arguments.

show abstract

Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy

Cited by 196 publications

References 5 publications

On linear instability of solitary waves for the nonlinear Dirac equation

On linear instability of solitary waves for the nonlinear Dirac equation

Approximate analytic solutions to coupled nonlinear Dirac equations

A variational study of some hadron bag models

Contact Info

Product

Resources

About