Communications in Mathematical Physics

, Volume 268, Issue 3, pp 757–817

Stable Directions for Small Nonlinear Dirac Standing Waves



We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schrödinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ABdMG96.
    Amrein W.O., Boutet de Monvel A., Georgescu V. (1996) C 0-groups, commutator methods and spectral theory of N-body Hamiltonians. Volume 135 of Progress in Mathematics. Birkhäuser Verlag, BaselGoogle Scholar
  2. Agm75.
    Agmon S. (1975) Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2(2): 151–218MATHMathSciNetGoogle Scholar
  3. AS86.
    Alvarez A., Soler M. (1986) Stability of the minimum solitary wave of a nonlinear spinorial model. Phys. Rev D 34: 644–645CrossRefADSGoogle Scholar
  4. BdMGS96.
    Boutet de Monvel A., Georgescu V., Sahbani J. (1996) Boundary values of resolvent families and propagation properties. C. R. Acad. Sci. Paris Sér. I Math. 322(3): 289–294MathSciNetGoogle Scholar
  5. BH92.
    Balslev E., Helffer B. (1992) Limiting absorption principle and resonances for the Dirac operator. Adv. Appl. Math. 13(2): 186–215MATHMathSciNetCrossRefGoogle Scholar
  6. BP92a.
    Buslaev V.S., Perel′man G.S.: Nonlinear scattering: states that are close to a soliton. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 200(Kraev. Zadachi Mat. Fiz. Smezh. Voprosy Teor. Funktsii. 24), 38–50, 70, 187 (1992)Google Scholar
  7. BP92b.
    Buslaev, V.S., Perel′man, G.S.: On nonlinear scattering of states which are close to a soliton. Méthodes semi-classiques, Vol. 2 (Nantes, (1991)), Astérisque 210(6), 49–63 (1992)Google Scholar
  8. BP92c.
    Buslaev V.S., Perel′man G.S. (1992) Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz 4(6): 63–102MATHMathSciNetGoogle Scholar
  9. BP95.
    Buslaev, V.S., G. S. Perel′man. On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear evolution equations. Volume 164 of Amer. Math. Soc. Transl. Ser. 2, Providence, RI: Amer. Math. Soc., 1995, pp. 75–98Google Scholar
  10. Bre77.
    Brenner P. (1977) L p-estimates of difference schemes for strictly hyperbolic systems with nonsmooth data. SIAM J. Numer. Anal. 14(6): 1126–1144MATHMathSciNetCrossRefGoogle Scholar
  11. Bre85.
    Brenner P. (1985) On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations. J. Differ. Eqs. 56(3): 310–344MATHMathSciNetCrossRefGoogle Scholar
  12. BS02.
    Buslaev, V.S., Sulem, C.: Asymptotic stability of solitary waves for nonlinear Schrödinger equations. In: The legacy of the inverse scattering transform in applied mathematics (South Hadley, MA, 2001) Vol. 301 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2002 pp. 163–181Google Scholar
  13. BS03.
    Buslaev V.S., Sulem C. (2003) On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(3): 419–475MATHMathSciNetCrossRefGoogle Scholar
  14. BSV87.
    Blanchard P., Stubbe J., Vàzquez L. (1987) Stability of nonlinear spinor fields with application to the Gross-Neveu model. Phys. Rev. D 36: 2422–2428CrossRefADSGoogle Scholar
  15. CF01.
    Cid C., Felmer P. (2001) Orbital stability and standing waves for the nonlinear Schrödinger equation with potential. Rev. Math. Phys. 13(12): 1529–1546MATHMathSciNetCrossRefGoogle Scholar
  16. CL82.
    Cazenave T., Lions P.-L. (1982) Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4): 549–561MATHMathSciNetCrossRefADSGoogle Scholar
  17. CS01.
    Cuccagna S., Schirmer P.P. (2001) On the wave equation with a magnetic potential. Comm. Pure Appl. Math. 54(2): 135–152MATHMathSciNetCrossRefGoogle Scholar
  18. Cuc01.
    Cuccagna S. (2001) Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54(9): 1110–1145MATHMathSciNetCrossRefGoogle Scholar
  19. Cuc03.
    Cuccagna S. (2003) On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15(8): 877–903MATHMathSciNetCrossRefGoogle Scholar
  20. Cuc05.
    Cuccagna, S.: Erratum: “Stabilization of solutions to nonlinear Schrödinger equations” [Comm. Pure Appl. Math. 54(9), 1110–1145 (2001). Comm. Pure Appl. Math. 58(1), 147 (2005)Google Scholar
  21. DF.
    D’Ancona, P., Fanelli, L.: Decay estimates for the wave and dirac equations with a magnetic potential. To appear on Comm. Pure Appl. MathGoogle Scholar
  22. ES04.
    Erdoğan M.B., Schlag W. (2004) Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. I. Dyn. Partial Differ. Eq. 1(4): 359–379MATHGoogle Scholar
  23. EV97.
    Escobedo M., Vega L. (2004) A semilinear Dirac equation in H s(R 3) for s > 1. SIAM J. Math. Anal. 28(2): 338–362MathSciNetCrossRefGoogle Scholar
  24. FS04.
    Fournais S., Skibsted E. (2004) Zero energy asymptotics of the resolvent for a class of slowly decaying potentials. Math. Z. 248(3): 593–633MATHMathSciNetCrossRefGoogle Scholar
  25. GM01.
    Georgescu V., Măntoiu M. (2001) On the spectral theory of singular Dirac type Hamiltonians. J. Operator Theory 46(2): 289–321MATHMathSciNetGoogle Scholar
  26. GNT04.
    Gustafson, S., Nakanishi, K., Tsai, T.-P.: Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int. Math. Res. Not. 66, 3559–3584 (2004)Google Scholar
  27. GS04.
    Goldberg M., Schlag W. (2004) A limiting absorption principle for the three-dimensional Schrödinger equation with L p potentials. Int. Math. Res. Not. 75: 4049–4071MATHMathSciNetCrossRefGoogle Scholar
  28. GSS87.
    Grillakis M., Shatah J., Strauss W. (1987) Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1): 160–197MATHMathSciNetCrossRefGoogle Scholar
  29. His00.
    Hislop, P.D.: Exponential decay of two-body eigenfunctions: a review. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999) Volume 4 of Electron. J. Differ. Eq. Conf. pp 265–288 (electronic)Google Scholar
  30. HS00.
    Hunziker W., Sigal I.M. (1987) Time-dependent scattering theory of N-body quantum systems. Rev. Math. Phys. 12(8): 1033–1084MathSciNetCrossRefGoogle Scholar
  31. HSS99.
    Hunziker W., Sigal I.M., Soffer A. (1999) Minimal escape velocities. Comm. Partial Differ. Eqs. 24(11-12): 2279–2295MATHMathSciNetGoogle Scholar
  32. IM99.
    Iftimovici A., Mantoiu M. (1999) Limiting absorption principle at critical values for the Dirac operator. Lett. Math. Phys. 49(3): 235–243MATHMathSciNetCrossRefGoogle Scholar
  33. JK79.
    Jensen A., Kato T. (1979) Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3): 583–611MATHMathSciNetCrossRefGoogle Scholar
  34. JN01.
    Jensen A., Nenciu G. (2001) A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6): 717–754MATHMathSciNetCrossRefGoogle Scholar
  35. JN04.
    Jensen, A., Nenciu, G.: Erratum: “A unified approach to resolvent expansions at thresholds” [Rev. Math. Phys. 16(6), 717–754 (2001)]. Rev. Math. Phys. 16(5), 675–677 (2004)Google Scholar
  36. JSS91.
    Journé J.-L., Soffer A., Sogge C.D. (1991) Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44(5): 573–604MATHMathSciNetGoogle Scholar
  37. Kat66.
    Kato T. (1965/1966) Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162: 258–279MathSciNetCrossRefGoogle Scholar
  38. KS05.
    Krieger, J., Schlag, W.: Stable manifolds for all supercritical monic nls in one dimension. Preprint, 2005,, 2005Google Scholar
  39. MSW79.
    Marshall, B., Strauss, W., Wainger, S.: Estimates from L p to its dual for the Klein-Gordon equation. In: Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2 Proc. Sympos. Pure Math., XXXV, Part, Providence, R.I.: Amer. Math. Soc., 1979 pp. 175–177Google Scholar
  40. MSW80.
    Marshall B., Strauss W., Wainger S. (1980) L p − L q estimates for the Klein-Gordon equation. J. Math. Pures Appl. (9). 59(4): 417–440MATHMathSciNetGoogle Scholar
  41. Par90.
    Parisse B. (1990) Résonances paires pour l’opérateur de Dirac. C. R. Acad. Sci. Paris Sér. I Math. 310(5): 265–268MATHMathSciNetGoogle Scholar
  42. PW97.
    Pillet C.-A., Wayne C.E. (1997) Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Eqs. 141(2): 310–326MATHMathSciNetCrossRefGoogle Scholar
  43. Ran.
    Ranada, A.F.: Classical nonlinear dirac field models of extended particles. In: Quantum Theory, Groups, Fields and Particles. Volume 198, A. O. Barut, ed., Amsterdam: Reidel, 1983, pp. 271–291Google Scholar
  44. RS78.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], (1978)Google Scholar
  45. RS79.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1979Google Scholar
  46. RSS05a.
    Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of n-soliton states of nls. To appear in Comm. Pure and Appl. Math. (2005)Google Scholar
  47. RSS05b.
    Rodnianski I., Schlag W., Soffer A. (2005) Dispersive analysis of charge transfer models. Comm. Pure Appl. Math. 58(2): 149–216MATHMathSciNetCrossRefGoogle Scholar
  48. Sch04.
    Schlag, W.: Stable manifolds for an orbitally unstable nls. Preprint, (2004)Google Scholar
  49. Sch05.
    Schlag, W.: Dispersive estimates for schroedinger operators: a survey. Preprint, (2005)Google Scholar
  50. SS85.
    Shatah J., Strauss W. (1985) Instability of nonlinear bound states. Commun. Math. Phys. 100(2): 173–190MATHMathSciNetCrossRefADSGoogle Scholar
  51. SS98.
    Sigal I.M., Soffer A. Local decay and velocity bounds for quantum propagation. Preprint, (1998)Google Scholar
  52. SV86.
    Strauss W.A., Vázquez L. (1986) Stability under dilations of nonlinear spinor fields. Phys. Rev. D 34: 641–643MathSciNetCrossRefADSGoogle Scholar
  53. SW90.
    Soffer A., Weinstein M.I. (1990) Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133(1): 119–146MATHMathSciNetCrossRefADSGoogle Scholar
  54. SW92.
    Soffer A., Weinstein M.I. (1992) Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Eqs. 98(2): 376–390MATHMathSciNetCrossRefGoogle Scholar
  55. SW99.
    Soffer A., Weinstein M.I. (1999) Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1): 9–74MATHMathSciNetCrossRefGoogle Scholar
  56. SW04.
    Soffer A., Weinstein M.I. (2004) Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16(8): 977–1071MATHMathSciNetCrossRefGoogle Scholar
  57. SW05.
    Soffer A., Weinstein M.I. (2005) Theory of Nonlinear Dispersive Waves and Selection of the Ground State. Phys. Rev. Lett. 95(21): 213905CrossRefADSGoogle Scholar
  58. Tha92.
    Thaller B. (1992) The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, BerlinMATHGoogle Scholar
  59. Tsa03.
    Tsai T.-P. (2003) Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. J. Differ. Eqs. 192(1): 225–282MATHCrossRefGoogle Scholar
  60. TY02a.
    Tsai T.-P., Yau H.-T. (2002) Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Comm. Pure Appl. Math. 55(2): 153–216MATHMathSciNetCrossRefGoogle Scholar
  61. TY02b.
    Tsai T.-P., Yau H.-T. (2002) Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data. Adv. Theor. Math. Phys. 6(1): 107–139MATHMathSciNetGoogle Scholar
  62. TY02c.
    Tsai T.-P., Yau H.-T. (2002) Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31: 1629–1673MATHMathSciNetCrossRefGoogle Scholar
  63. TY02d.
    Tsai T.-P., Yau H.-T. (2002) Stable directions for excited states of nonlinear Schrödinger equations. Comm. Partial Differ. Eqs. 27(11-12): 2363–2402MATHMathSciNetCrossRefGoogle Scholar
  64. Wed00.
    Weder R. (2000) Center manifold for nonintegrable nonlinear Schrödinger equations on the line. Commun. Math. Phys. 215(2): 343–356MATHMathSciNetCrossRefADSGoogle Scholar
  65. Yaj95.
    Yajima K. (1995) The W k,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47(3): 551–581MATHMathSciNetGoogle Scholar
  66. Yam93.
    Yamada O. (1995) A remark on the limiting absorption method for Dirac operators. Proc. Japan Acad. Ser. A Math. Sci. 69(7): 243–246CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.CeremadeUniversité Paris DauphineParis Cédex 16France

Personalised recommendations