On scattering of solitons for the Klein-Gordon equation coupled to a particle. (English) Zbl 1127.35054

Summary: We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein-Gordon equation coupled to a charged relativistic particle. The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Klein-Gordon equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the “Fermi golden rule”. The proof is based on an extension of the general strategy introduced by Soffer and Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.


35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35P25 Scattering theory for PDEs
35Q51 Soliton equations
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI arXiv


[1] Abraham M., (1905) Theorie der Elektrizitat, Band 2: Elektromagnetische Theorie der Strahlung. Leipzig, Teubner
[2] Agmon S. (1975) Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Ser. 2(IV): 151–218 · Zbl 0315.47007
[3] Arnold V.I., Kozlov V.V., Neishtadt A.I., (1997) Mathematical Aspects of Classical and Celestial Mechanics. Berlin, Springer · Zbl 0885.70001
[4] Beresticky H., Lions P.L. (1983) Nonlinear scalar field equations. Arch. Rat. Mech. Anal. 82(4): 313–375
[5] Buslaev V.S., Perelman G.S.: On nonlinear scattering of states which are close to a soliton. In: Méthodes Semi-Classiques, Vol. 2 Colloque International (Nantes, Juin 1991), Asterisque 208 (1992), pp. 49–63 · Zbl 0795.35111
[6] Buslaev V.S., Perelman G.S. (1993) Scattering for the nonlinear Schrödinger equation: states close to a soliton. St. Petersburg Math. J. 4: 1111–1142
[7] Buslaev V.S., Perelman G.S. (1995) On the stability of solitary waves for nonlinear Schrödinger equations. Trans. Amer. Math. Soc. 164, 75–98 · Zbl 0841.35108
[8] Buslaev V.S., Sulem C. (2003) On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20(3): 419–475 · Zbl 1028.35139
[9] Cuccagna S. (2001) Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54 (9): 1110–1145 · Zbl 1031.35129
[10] Cuccagna S. (2003) On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15(8): 877–903 · Zbl 1084.35089
[11] Dirac P.A.M. (1938) Classical theory of radiating electrons. Proc. Roy. Soc. (London) A 167, 148–169 · Zbl 0023.42702
[12] Eckhaus W., van Harten A.: The Inverse Scattering Transformation and the Theory of Solitons. An Introduction. Amsterdam: North-Holland, 1981 · Zbl 0463.35001
[13] Esteban M., Georgiev V., Sere E. (1996) Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations. Calc. Var. Partial Differ. Equ. 4(3): 265–281 · Zbl 0869.35105
[14] Grillakis M., Shatah J., Strauss W.A.: Stability theory of solitary waves in the presence of symmetry I, II. J. Func. Anal. 74, 160–197 (1987); 94, 308–348 (1990) · Zbl 0656.35122
[15] Imaikin V., Komech A., Markowich P. (2003) Scattering of solitons of the Klein–Gordon equation coupled to a classical particle. J. Math. Phys. 44(3): 1202–1217 · Zbl 1061.35070
[16] Imaikin V., Komech A., Mauser N. (2004) Soliton-type asymptotics for the coupled Maxwell–Lorentz equations. Ann. Inst. Poincaré, Phys. Theor. 5, 1117–1135 · Zbl 1067.35132
[17] Imaikin V., Komech A., Spohn H. (2002) Soliton-like asymptotics and scattering for a particle coupled to Maxwell field. Russ. J. Math. Phys. 9(4): 428–436 · Zbl 1104.78301
[18] Imaikin V., Komech A., Spohn H. (2003) Scattering theory for a particle coupled to a scalar field. J. Disc. Cont. Dyn. Sys. 10(1–2): 387–396 · Zbl 1052.37057
[19] Imaikin V., Komech A., Spohn H. (2004) Rotating charge coupled to the Maxwell field: scattering theory and adiabatic limit. Monatsh. Math. 142(1–2): 143–156 · Zbl 1082.35145
[20] Jensen A. (2001) On a unified approach to resolvent expansions for Schrödinger operators. RIMS Kokyuroku 1208, 91–103 · Zbl 0991.35502
[21] Jensen A., Kato T. (1979) Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 · Zbl 0448.35080
[22] Komech A.I.: Linear Partial Differential Equations with Constant Coefficients. In: Yu.V. Egorov A.I. Komech, M.A. Shubin, Elements of the Modern Theory of Partial Differential Equations, Berlin: Springer, 1999, pp.127–260
[23] Komech A., Kunze M., Spohn H. (1999) Effective dynamics for a mechanical particle coupled to a wave field. Commun. Math. Phys. 203, 1–19 · Zbl 0947.70010
[24] Komech A., Kunze M., Spohn H. (1997) Long-time asymptotics for a classical particle interacting with a scalar wave field. Comm. Part. Differ. Eqs. 22, 307–335 · Zbl 0878.35094
[25] Komech A.I., Spohn H. (1998) Soliton-like asymptotics for a classical particle interacting with a scalar wave field. Nonlin. Analysis 33, 13–24 · Zbl 0935.37046
[26] Lorentz H.A.: Theory of Electrons. 2nd edition 1915. Reprinted by New York, Dover, 1952 · Zbl 0048.35205
[27] Miller J., Weinstein M. (1996) Asymptotic stability of solitary waves for the regularized long-wave equation. Comm. Pure Appl. Math. 49(4): 399–441 · Zbl 0854.35102
[28] Pego R.L., Weinstein M.I. (1992) On asymptotic stability of solitary waves. Phys. Lett. A 162, 263–268
[29] Pego R.L., Weinstein M.I. (1994) Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 · Zbl 0805.35117
[30] Sigal I.M. (1993) Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2): 297–320 · Zbl 0780.35106
[31] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable systems. In: Proceedings of Conference on an Integrable and Nonintegrable Systems, June, 1988, Oleron, France, Integrable Systems and Applications, Springer Lecture Notes in Physics, Vol. 342, Berlin-Heidelberg-New York: Springer, 1989 · Zbl 0721.35082
[32] Soffer A., Weinstein M.I. (1990) Multichannel nonlinear scattering in nonintegrable systems. Commun. Math. Phys. 133, 119–146 · Zbl 0721.35082
[33] Soffer A., Weinstein M.I. (1992) Multichannel nonlinear scattering and stability II The case of Anisotropic and potential and data. J. Differ. Eqs. 98, 376–390 · Zbl 0795.35073
[34] Soffer A., Weinstein M.I. (1999) Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1): 9–74 · Zbl 0910.35107
[35] Soffer A., Weinstein M.I. (2004) Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16(8): 977–1071 · Zbl 1111.81313
[36] Soffer A., Weinstein M.I. (2005) Theory of nonlinear dispersive waves and selection of the ground state. Phys. Rev. Lett. 95: 213905
[37] Spohn H., (2004) Dynamics of Charged Particles and Their Radiation Field. Cambridge, Cambridge University Press · Zbl 1078.81004
[38] Vainberg B.: Behavior of the solution of the Cauchy problem for a hyperbolic equation as t Math. of the USSR – Sbornik 7(4), 533–568 (1969); trans. Mat. Sb. 78(4), 542–578 (1969)
[39] Vainberg B. (1975) On the short wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as t of solutions of non-stationary problems. Russ. Math. Surv. 30(2): 1–58 · Zbl 0318.35006
[40] Vainberg B., (1989) Asymptotic methods in equations of mathematical physics. New York–London, Gordon and Breach Publishers · Zbl 0743.35001
[41] Weinstein M. (1985) Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3): 472–491 · Zbl 0583.35028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.