Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abraham M., (1905) Theorie der Elektrizitat, Band 2: Elektromagnetische Theorie der Strahlung. Leipzig, Teubner
Agmon S. (1975) Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Ser. 2(IV): 151–218
Arnold V.I., Kozlov V.V., Neishtadt A.I., (1997) Mathematical Aspects of Classical and Celestial Mechanics. Berlin, Springer
Beresticky H., Lions P.L. (1983) Nonlinear scalar field equations. Arch. Rat. Mech. Anal. 82(4): 313–375
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
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
Buslaev V.S., Perelman G.S. (1995) On the stability of solitary waves for nonlinear Schrödinger equations. Trans. Amer. Math. Soc. 164, 75–98
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
Cuccagna S. (2001) Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54 (9): 1110–1145
Cuccagna S. (2003) On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15(8): 877–903
Dirac P.A.M. (1938) Classical theory of radiating electrons. Proc. Roy. Soc. (London) A 167, 148–169
Eckhaus W., van Harten A.: The Inverse Scattering Transformation and the Theory of Solitons. An Introduction. Amsterdam: North-Holland, 1981
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
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)
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
Imaikin V., Komech A., Mauser N. (2004) Soliton-type asymptotics for the coupled Maxwell–Lorentz equations. Ann. Inst. Poincaré, Phys. Theor. 5, 1117–1135
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
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
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
Jensen A. (2001) On a unified approach to resolvent expansions for Schrödinger operators. RIMS Kokyuroku 1208, 91–103
Jensen A., Kato T. (1979) Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611
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
Komech A., Kunze M., Spohn H. (1999) Effective dynamics for a mechanical particle coupled to a wave field. Commun. Math. Phys. 203, 1–19
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
Komech A.I., Spohn H. (1998) Soliton-like asymptotics for a classical particle interacting with a scalar wave field. Nonlin. Analysis 33, 13–24
Lorentz H.A.: Theory of Electrons. 2nd edition 1915. Reprinted by New York, Dover, 1952
Miller J., Weinstein M. (1996) Asymptotic stability of solitary waves for the regularized long-wave equation. Comm. Pure Appl. Math. 49(4): 399–441
Pego R.L., Weinstein M.I. (1992) On asymptotic stability of solitary waves. Phys. Lett. A 162, 263–268
Pego R.L., Weinstein M.I. (1994) Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349
Sigal I.M. (1993) Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2): 297–320
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
Soffer A., Weinstein M.I. (1990) Multichannel nonlinear scattering in nonintegrable systems. Commun. Math. Phys. 133, 119–146
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
Soffer A., Weinstein M.I. (1999) Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1): 9–74
Soffer A., Weinstein M.I. (2004) Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16(8): 977–1071
Soffer A., Weinstein M.I. (2005) Theory of nonlinear dispersive waves and selection of the ground state. Phys. Rev. Lett. 95: 213905
Spohn H., (2004) Dynamics of Charged Particles and Their Radiation Field. Cambridge, Cambridge University Press
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)
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
Vainberg B., (1989) Asymptotic methods in equations of mathematical physics. New York–London, Gordon and Breach Publishers
Weinstein M. (1985) Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3): 472–491
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Supported partly by Austrian Science Foundation (FWF) Project P19138-N13, by research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).
On leave Department Mechanics and Mathematics of Moscow State University. Supported partly by Austrian Science Foundation (FWF) Project P19138-N13 by Max-Planck Institute of Mathematics in the Sciences (Leipzig), and Wolfgang Pauli Institute of Vienna University.
Supported partially by the NSF grant DMS-0405927
Rights and permissions
About this article
Cite this article
Imaikin, V., Komech, A. & Vainberg, B. On Scattering of Solitons for the Klein–Gordon Equation Coupled to a Particle. Commun. Math. Phys. 268, 321–367 (2006). https://doi.org/10.1007/s00220-006-0088-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0088-z