Geodesic
On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D
G. Berkolaiko1 ; A. Comech1, 2
1 Mathematics Department, Texas A&M University, College Station, TX 77843, USA
2 Institute for Information Transmission Problems, Moscow 101447, Russia
Mathematical modelling of natural phenomena, Tome 7 (2012) no. 2, pp. 13-31.

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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.
DOI : 10.1051/mmnp/20127202

G. Berkolaiko 1 ; A. Comech 1, 2

1 Mathematics Department, Texas A&M University, College Station, TX 77843, USA
2 Institute for Information Transmission Problems, Moscow 101447, Russia 
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G. Berkolaiko; A. Comech. On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D. Mathematical modelling of natural phenomena, Tome 7 (2012) no. 2, pp. 13-31. doi : 10.1051/mmnp/20127202. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20127202/

[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 St. Petersburg Math. J. 1993 1111 1142

[3] V. S. Buslaev, C. Sulem Ann. Inst. H. Poincaré Anal. Non Linéaire 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 Comm. Math. Phys. 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 Comm. Pure Appl. Math. 2001 1110 1145

[8] T. Cazenave, L. Vázquez Comm. Math. Phys. 1986 35 47

[9] G. H. Derrick J. Mathematical Phys. 1964 1252 1254

[10] D. J. Gross, A. Neveu Phys. Rev. D 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 Comm. Pure Appl. Math. 1988 747 774

[13] L. Gross Comm. Pure Appl. Math. 1966 1 15

[14] M. Grillakis, J. Shatah, W. Strauss J. Funct. Anal. 1987 160 197

[15] S. Y. Lee, A. Gavrielides Phys. Rev. D 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 Comm. Math. Phys. 1983 313 327

[18] J. Shatah Trans. Amer. Math. Soc. 1985 701 710

[19] M. Soler Phys. Rev. D 1970 2766 2769

[20] J. Shatah, W. Strauss Comm. Math. Phys. 1985 173 190

[21] A. Soffer, M. I. Weinstein J. Differential Equations 1992 376 390

[22] A. Soffer, M. I. Weinstein Invent. Math. 1999 9 74

[23] N. G. Vakhitov, A. A. Kolokolov Radiophys. Quantum Electron. 1973 783 789

[24] M. I. Weinstein SIAM J. Math. Anal. 1985 472 491

[25] M. I. Weinstein Comm. Pure Appl. Math. 1986 51 67

[26] V. Zakharov Zh. Éksp. Teor. Fiz 1967 1735 1743

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