We consider the nonlinear Dirac equation, also known as the Soler model:
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.
@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}, pages = {639--654}, publisher = {Elsevier}, volume = {31}, number = {3}, year = {2014}, doi = {10.1016/j.anihpc.2013.06.001}, mrnumber = {3208458}, zbl = {1297.35029}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.001/} }
TY - JOUR AU - Comech, Andrew AU - Guan, Meijiao AU - Gustafson, Stephen TI - On linear instability of solitary waves for the nonlinear Dirac equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 639 EP - 654 VL - 31 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.001/ DO - 10.1016/j.anihpc.2013.06.001 LA - en ID - AIHPC_2014__31_3_639_0 ER -
%0 Journal Article %A Comech, Andrew %A Guan, Meijiao %A Gustafson, Stephen %T On linear instability of solitary waves for the nonlinear Dirac equation %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 639-654 %V 31 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.001/ %R 10.1016/j.anihpc.2013.06.001 %G en %F 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, Volume 31 (2014) no. 3, pp. 639-654. doi : 10.1016/j.anihpc.2013.06.001. http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.001/
[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 | EuDML | MR | Zbl
, ,[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 | Zbl
, ,[6] Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313 -345 | MR | Zbl
, ,[7] On spinor soliton stability, Phys. Lett. A 73 (1979), 87 -90 | MR
,[8] Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys. 268 (2006), 757 -817 | MR | Zbl
,[9] On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case, SIAM J. Math. Anal. 40 (2008), 1621 -1670 | MR | Zbl
,[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
, , , ,[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 | Zbl
, ,[14] Block-diagonalization of the symmetric first-order coupled-mode system, SIAM J. Appl. Dyn. Syst. 5 (2006), 66 -83 | MR | Zbl
, ,[15] On instability of excited states of the nonlinear Schrödinger equation, Phys. D 238 (2009), 38 -54 | MR | Zbl
,[16] On the Darboux and Birkhoff steps in the asymptotic stability of solitons, arXiv:1203.2120 (2012) | MR | Zbl
,[17] Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys. 105 (1986), 35 -47 | MR | Zbl
, ,[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 | Zbl
, , ,[20] Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys. 171 (1995), 323 -350 | MR | Zbl
, ,[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 | Zbl
, ,[23] The Cauchy problem for the coupled Maxwell and Dirac equations, Comm. Pure Appl. Math. 19 (1966), 1 -15 | MR | Zbl
,[24] Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160 -197 | MR | Zbl
, , ,[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 | Zbl
, ,[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 | Zbl
,[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 | EuDML | Numdam | MR | Zbl
,[31] On the eigenfunctions of the equation , Dokl. Akad. Nauk SSSR 165 (1965), 36 -39 | MR
,[32] Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys. 53 (2012), 073 -705 | MR
, ,[33] Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978) | MR | Zbl
, ,[34] Solitons and Particles: Papers, World Scientific Pub. (1984) | MR
, ,[35] Stable standing waves of nonlinear Klein–Gordon equations, Comm. Math. Phys. 91 (1983), 313 -327 | MR | Zbl
,[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 | Zbl
, ,[38] Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 -162 | MR | Zbl
,[39] Stability under dilations of nonlinear spinor fields, Phys. Rev. D (3) 34 (1986), 641 -643 | MR | Zbl
, ,[40] The Dirac Equation, Texts Monogr. Phys. , Springer-Verlag, Berlin (1992) | MR | Zbl
,[41] A soluble relativistic field theory, Ann. Physics 3 (1958), 91 -112 | MR | Zbl
,[42] Die gruppentheoretische Methode in der Quantenmechanik, Springer-Verlag, Berlin (1932) | Zbl
,[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 | Zbl
,Cited by Sources: