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

G. Berkolaiko; A. Comech

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 13-31
  • ISSN: 0973-5348

Abstract

top
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.

How to cite

top

Berkolaiko, G., and Comech, A.. "On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D⋆⋆." Mathematical Modelling of Natural Phenomena 7.2 (2012): 13-31. <http://eudml.org/doc/222290>.

@article{Berkolaiko2012,
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.},
author = {Berkolaiko, G., Comech, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {nonlinear Dirac equation; Dirac operator; spectral stability; linear instability; solitary waves; Soler model; massive Gross-Neveu model; Jost solution; Evans function},
language = {eng},
month = {2},
number = {2},
pages = {13-31},
publisher = {EDP Sciences},
title = {On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D⋆⋆},
url = {http://eudml.org/doc/222290},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Berkolaiko, G.
AU - Comech, A.
TI - On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D⋆⋆
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 13
EP - 31
AB - 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.
LA - eng
KW - nonlinear Dirac equation; Dirac operator; spectral stability; linear instability; solitary waves; Soler model; massive Gross-Neveu model; Jost solution; Evans function
UR - http://eudml.org/doc/222290
ER -

References

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.