Nous considérons l'équation de Dirac non linéaire, aussi connue comme modèle de Soler. Nous étudions le spectre ponctuel des linéarisations autour d'ondes solitaires de petite amplitude dans la limite , et montrons que si une valeur propre positive et une négative sont présentes dans le spectre des linéarisations autour de ces ondes solitaires lorsque ω est suffisamment proche de m, ce qui entraîne que ces ondes solitaires sont linéairement instables. L'approche est basée sur l'application de la théorie des perturbations 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.
We consider the nonlinear Dirac equation, also known as the Soler model: We study the point spectrum of linearizations at small amplitude solitary waves in the limit , proving that if , 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.
@article{AIHPC_2014__31_3_639_0, author = {Comech, Andrew and Guan, Meijiao and Gustafson, Stephen}, title = {On linear instability of solitary waves for the nonlinear Dirac equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {31}, year = {2014}, pages = {639-654}, doi = {10.1016/j.anihpc.2013.06.001}, mrnumber = {3208458}, zbl = {1297.35029}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_3_639_0} }
Comech, Andrew; Guan, Meijiao; Gustafson, Stephen. On linear instability of solitary waves for the nonlinear Dirac equation. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 639-654. doi : 10.1016/j.anihpc.2013.06.001. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_3_639_0/
[1] Energetic stability criterion for a nonlinear spinorial model, Phys. Rev. Lett. 50 (1983), 1230 -1233
, ,[2] Stability of the minimum solitary wave of a nonlinear spinorial model, Phys. Rev. D 34 (1986), 644 -645
, ,[3] On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom. 7 (2012), 13 -31 | MR 2892774 | Zbl 1247.35117
, ,[4] On spectral stability of nonlinear Dirac equation, arXiv:1211.3336 (2012)
, ,[5] On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations 37 (2012), 1001 -1056 | MR 2924465 | Zbl 1251.35098
, ,[6] Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313 -345 | MR 695535 | Zbl 0533.35029
, ,[7] On spinor soliton stability, Phys. Lett. A 73 (1979), 87 -90 | MR 592382
,[8] Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys. 268 (2006), 757 -817 | MR 2259214 | Zbl 1127.35060
,[9] On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case, SIAM J. Math. Anal. 40 (2008), 1621 -1670 | MR 2466169 | Zbl 1167.35438
,[10] Spectral stability of nonlinear waves in dynamical systems, McMaster University, Hamilton, Ontario, Canada (2007)
,[11] Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity, Phys. Rev. E 82 (2010), 036604 | MR 2788016
, , , ,[12] On the meaning of the Vakhitov–Kolokolov stability criterion for the nonlinear Dirac equation, arXiv:1107.1763 (2011)
,[13] Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56 (2003), 1565 -1607 | MR 1995870 | Zbl 1072.35165
, ,[14] Block-diagonalization of the symmetric first-order coupled-mode system, SIAM J. Appl. Dyn. Syst. 5 (2006), 66 -83 | MR 2217129 | Zbl 1102.35062
, ,[15] On instability of excited states of the nonlinear Schrödinger equation, Phys. D 238 (2009), 38 -54 | MR 2571965 | Zbl 1161.35500
,[16] On the Darboux and Birkhoff steps in the asymptotic stability of solitons, arXiv:1203.2120 (2012) | MR 3019561 | Zbl 06151718
,[17] Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys. 105 (1986), 35 -47 | MR 847126 | Zbl 0596.35117
, ,[18] The quantum theory of the electron, Proc. Roy. Soc. A. 117 (1928), 616 -624
,[19] Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N. S.) 45 (2008), 535 -593 | MR 2434346 | Zbl 1288.49016
, , ,[20] Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys. 171 (1995), 323 -350 | MR 1344729 | Zbl 0843.35114
, ,[21] Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974), 3235 -3253
, ,[22] Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations, J. Math. Soc. Japan 64 (2012), 533 -548 | MR 2916078 | Zbl 1253.35158
, ,[23] The Cauchy problem for the coupled Maxwell and Dirac equations, Comm. Pure Appl. Math. 19 (1966), 1 -15 | MR 190520 | Zbl 0137.32401
,[24] Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160 -197 | MR 901236 | Zbl 0656.35122
, , ,[25] Solitary wave solutions for the nonlinear Dirac equations, arXiv:0812.2273 (2008)
,[26] The nonlinear Dirac equation in Bose–Einstein condensates: foundation and symmetries, Phys. D 238 (2009), 1413 -1421 | MR 2542748 | Zbl 1167.82306
, ,[27] Quantization of the localized solutions in two-dimensional field theories of massive fermions, Phys. Rev. D 12 (1975), 3880 -3886
, ,[28] Existence of stationary states for nonlinear Dirac equations, J. Differential Equations 74 (1988), 50 -68 | MR 949625 | Zbl 0696.35154
,[29] Chiral confinement in quasirelativistic Bose–Einstein condensates, Phys. Rev. Lett. 104 (2010), 073603
, , , , ,[30] Contributions mathématiques à la théorie des matrices de Dirac, Ann. Inst. H. Poincaré 6 (1936), 109 -136 | Numdam | MR 1508031 | Zbl 0015.19403
,[31] On the eigenfunctions of the equation , Dokl. Akad. Nauk SSSR 165 (1965), 36 -39 | MR 192184
,[32] Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys. 53 (2012), 073 -705 | MR 2985264
, ,[33] Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978) | MR 493421 | Zbl 0401.47001
, ,[34] Solitons and Particles: Papers, World Scientific Pub. (1984) | MR 802075
, ,[35] Stable standing waves of nonlinear Klein–Gordon equations, Comm. Math. Phys. 91 (1983), 313 -327 | MR 723756 | Zbl 0539.35067
,[36] Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D 1 (1970), 2766 -2769
,[37] Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173 -190 | MR 804458 | Zbl 0603.35007
, ,[38] Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 -162 | MR 454365 | Zbl 0356.35028
,[39] Stability under dilations of nonlinear spinor fields, Phys. Rev. D (3) 34 (1986), 641 -643 | MR 848095 | Zbl 1222.81167
, ,[40] The Dirac Equation, Texts Monogr. Phys. , Springer-Verlag, Berlin (1992) | MR 1219537 | Zbl 0881.47021
,[41] A soluble relativistic field theory, Ann. Physics 3 (1958), 91 -112 | MR 91788 | Zbl 0078.44303
,[42] Die gruppentheoretische Methode in der Quantenmechanik, Springer-Verlag, Berlin (1932) | Zbl 0004.08905
,[43] Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron. 16 (1973), 783 -789
, ,[44] Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472 -491 | MR 783974 | Zbl 0583.35028
,