Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T22:12:52.276Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 February 2012

G. Berkolaiko  and
A. Comech
G. Berkolaiko
Affiliation:
Mathematics Department, Texas A&M University, College Station, TX 77843, USA
A. Comech*
Affiliation:
Mathematics Department, Texas A&M University, College Station, TX 77843, USA Institute for Information Transmission Problems, Moscow 101447, Russia
*
⋆ Corresponding author. E-mail: comech@math.tamu.edu
Get access

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.

Keywords

nonlinear Dirac equationDirac operatorspectral stabilitylinear instabilitysolitary wavesSoler modelmassive Gross-Neveu modelJost solutionEvans function
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 13 - 31
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

N. Boussaid, S. Cuccagna. On stability of standing waves of nonlinear Dirac equations. ArXiv e-prints 1103.4452, (2011).
Buslaev, V. S., Perel'an, G. S.. Scattering for the nonlinear Schrödinger equation : states that are close to a soliton. St. Petersburg Math. J., 4 (1993), 1111–1142. Google Scholar
Buslaev, V. S., Sulem, C.. On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 419–475. CrossRefGoogle Scholar
M. Chugunova. Spectral stability of nonlinear waves in dynamical systems (Doctoral Thesis). McMaster University, Hamilton, Ontario, Canada, 2007.
Cuccagna, S., Mizumachi, T.. On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Comm. Math. Phys., 284 (2008), 51–77. CrossRefGoogle Scholar
A. Comech. On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation. ArXiv e-prints, (2011), arXiv :1107.1763.
Cuccagna, S.. Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math., 54 (2001), 1110–1145. CrossRefGoogle Scholar
Cazenave, T., Vázquez, L.. Existence of localized solutions for a classical nonlinear Dirac field. Comm. Math. Phys., 105 (1986), 35–47. CrossRefGoogle Scholar
Derrick, G. H.. Comments on nonlinear wave equations as models for elementary particles. J. Mathematical Phys., 5 (1964), 1252–1254. CrossRefGoogle Scholar
Gross, D. J., Neveu, A.. Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D, 10 (1974), 3235–3253. CrossRefGoogle Scholar
V. Georgiev, M. Ohta. Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations. ArXiv e-prints, (2010).
Grillakis, M.. Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Comm. Pure Appl. Math., 41 (1988), 747–774. CrossRefGoogle Scholar
Gross, L.. The Cauchy problem for the coupled Maxwell and Dirac equations. Comm. Pure Appl. Math., 19 (1966), 1–15. CrossRefGoogle Scholar
Grillakis, M., Shatah, J., Strauss, W.. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal., 74 (1987), 160–197. CrossRefGoogle Scholar
Lee, S. Y., Gavrielides, A.. Quantization of the localized solutions in two-dimensional field theories of massive fermions. Phys. Rev. D, 12 (1975), 3880–3886. CrossRefGoogle Scholar
D. E. Pelinovsky, A. Stefanov. Asymptotic stability of small gap solitons in the nonlinear Dirac equations. ArXiv e-prints, (2010), arXiv :1008.4514.
Shatah, J.. Stable standing waves of nonlinear Klein-Gordon equations. Comm. Math. Phys., 91 (1983), 313–327. CrossRefGoogle Scholar
Shatah, J.. Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math. Soc., 290 (1985), 701–710. CrossRefGoogle Scholar
Soler, M.. Classical, stable, nonlinear spinor field with positive rest energy. Phys. Rev. D, 1 (1970), 2766–2769. CrossRefGoogle Scholar
Shatah, J., Strauss, W.. Instability of nonlinear bound states. Comm. Math. Phys., 100 (1985), 173–190. CrossRefGoogle Scholar
Soffer, A., Weinstein, M. I.. Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differential Equations, 98 (1992), 376–390. CrossRefGoogle Scholar
Soffer, A., Weinstein, M. I.. Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math., 136 (1999), 9–74. CrossRefGoogle Scholar
Vakhitov, N. G., Kolokolov, A. A.. Stationary solutions of the wave equation in the medium with nonlinearity saturation. Radiophys. Quantum Electron., 16 (1973), 783–789. CrossRefGoogle Scholar
Weinstein, M. I.. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal., 16 (1985), 472–491. CrossRefGoogle Scholar
Weinstein, M. I.. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math., 39 (1986), 51–67. CrossRefGoogle Scholar
Zakharov, V.. Instability of self-focusing of light. Zh. Éksp. Teor. Fiz, 53 (1967), 1735–1743. Google Scholar