Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 010, 23 pages      arXiv:0711.0041      https://doi.org/10.3842/SIGMA.2008.010
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation

Alexander I. Komech a, c and Andrew A. Komech b, c
a) Faculty of Mathematics, University of Vienna, Wien A-1090, Austria
b) Mathematics Department, Texas A&M University, College Station, TX 77843, USA
c) Institute for Information Transmission Problems, B. Karetny 19, Moscow 101447, Russia

Received November 01, 2007, in final form January 22, 2008; Published online January 31, 2008

Abstract
We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give examples of systems which contain such solutions.

Key words: global attractors; solitary waves; solitary asymptotics; nonlinear Klein-Gordon equation; dispersive Hamiltonian systems; unitary invariance.

pdf (449 kb)   ps (319 kb)   tex (171 kb)

References

  1. Babin A.V., Vishik M.I., Attractors of evolution equations, of Studies in Mathematics and its Applications, Vol. 25, North-Holland Publishing Co., Amsterdam, 1992 (translated and revised from the 1989 Russian original by Babin).
  2. Berestycki H., Lions P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345.
  3. Berestycki H., Lions P.-L., Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), 347-375.
  4. Bohr N., On the constitution of atoms and molecules, Phil. Mag. 26 (1913), 1-25.
  5. Brezis H., Lieb E.H., Minimum action solutions of some vector field equations, Comm. Math. Phys. 96 (1984), 97-113.
  6. Broglie L.D., Recherches sur la théorie des Quanta, Thèses, Paris, 1924.
  7. Buslaev V.S., Perel'man G.S., Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, St. Petersburg Math. J. 4 (1993), 1111-1142.
  8. Buslaev V.S., Perel'man G.S., On the stability of solitary waves for nonlinear Schrödinger equations, in Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, Vol. 164, Amer. Math. Soc., Providence, RI, 1995, 75-98.
  9. Buslaev V.S., Sulem C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 419-475.
  10. Cazenave T., Vázquez L., Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys. 105 (1986), 35-47.
  11. Comech A.A., Numerical simulations of the Klein-Gordon field with nonlinear interaction (2007), scripts for GNU Octave, http://www.math.tamu.edu/~comech/tools/kg-string.
  12. Cuccagna S., Asymptotic stability of the ground states of the nonlinear Schrödinger equation, Rend. Istit. Mat. Univ. Trieste 32 (2001), 105-118.
  13. Cuccagna S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), 1110-1145.
  14. Cuccagna S., On asymptotic stability of ground states of NLS, Rev. Math. Phys. 15 (2003), 877-903.
  15. Derrick G.H., Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. 5 (1964), 1252-1254.
  16. Esteban M.J., Georgiev V., Séré E., Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations 4 (1996), 265-281.
  17. Esteban M.J., Séré É., Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys. 171 (1995), 323-350.
  18. Gell-Mann M., Ne'eman Y., The eightfold way, W.A. Benjamin, Inc., New York, NY, 1964.
  19. Ginibre J., Velo G., Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), 399-442.
  20. Glassey R.T., Strauss W.A., Decay of a Yang-Mills field coupled to a scalar field, Comm. Math. Phys. 67 (1979), 51-67.
  21. Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160-197.
  22. Guo Y., Nakamitsu K., Strauss W., Global finite-energy solutions of the Maxwell-Schrödinger system, Comm. Math. Phys. 170 (1995), 181-196.
  23. Henry D., Geometric theory of semilinear parabolic equations, Springer, 1981.
  24. Hörmander L., The analysis of linear partial differential operators. I, 2nd ed., Springer Study Edition, Springer-Verlag, Berlin, 1990.
  25. Hörmander L., On the fully nonlinear Cauchy problem with small data. II, in Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988-1989), IMA Vol. Math. Appl., Vol. 30, Springer, New York, 1991, 51-81.
  26. Jörgens K., Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295-308.
  27. Klainerman S., Long-time behavior of solutions to nonlinear evolution equations, Arch. Ration. Mech. Anal. 78 (1982), 73-98.
  28. Komech A., On transitions to stationary states in one-dimensional nonlinear wave equations, Arch. Ration. Mech. Anal. 149 (1999), 213-228.
  29. Komech A., Spohn H., Long-time asymptotics for the coupled Maxwell-Lorentz equations, Comm. Partial Differential Equations 25 (2000), 559-584.
  30. Komech A., Spohn H., Kunze M., Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Differential Equations 22 (1997), 307-335.
  31. Komech A., Vainberg B., On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations, Arch. Ration. Mech. Anal. 134 (1996), 227-248.
  32. Komech A.I., Stabilization of the interaction of a string with a nonlinear oscillator, Mosc. Univ. Math. Bull. 46 (1991), 34-39.
  33. Komech A.I., On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appl. 196 (1995), 384-409.
  34. Komech A.I., On attractor of a singular nonlinear U(1)-invariant Klein-Gordon equation, in Progress in analysis, Vol. I, II (2001, Berlin), World Sci. Publishing, River Edge, NJ, 2003, 599-611.
  35. Komech A.I., Komech A.A., Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal. 185 (2007), 105-142, math.AP/0609013.
  36. Komech A.I., Komech A.A., On global attraction to quantum stationary states II. Several nonlinear oscillators coupled to massive scalar field, MPI Preprint Series 17/2007 (2007), available at http://www.mis.mpg.de/preprints/2007/prepr2007_17.html.
  37. Komech A.I., Komech A.A., On global attraction to quantum stationary states III. Klein-Gordon equation with mean field interaction, MPI Preprint Series 66/2007 (2007), available at http://www.mis.mpg.de/preprints/2007/prepr2007_66.html.
  38. Levin B.Y., Lectures on entire functions, Translations of Mathematical Monographs, Vol. 150, American Mathematical Society, Providence, RI, 1996 (in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, translated from the Russian manuscript by Tkachenko).
  39. Morawetz C.S., Strauss W.A., Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1-31.
  40. Pillet C.-A., Wayne C.E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differential Equations 141 (1997), 310-326.
  41. Schiff L.I., Nonlinear meson theory of nuclear forces. I. Neutral scalar mesons with point-contact repulsion, Phys. Rev. 84 (1951), 1-9.
  42. Schiff L.I., Nonlinear meson theory of nuclear forces. II. Nonlinearity in the meson-nucleon coupling, Phys. Rev. 84 (1951), 10-11.
  43. Schrödinger E., Quantisierung als eigenwertproblem, Ann. Phys. 81 (1926), 109-139.
  44. Segal I.E., The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France 91 (1963), 129-135.
  45. Segal I.E., Non-linear semi-groups, Ann. of Math. (2) 78 (1963), 339-364.
  46. Segal I.E., Quantization and dispersion for nonlinear relativistic equations, in Proc. Conf. "Mathematical Theory of Elementary Particles" (1965, Dedham, Mass.), M.I.T. Press, Cambridge, Mass., 1966, 79-108.
  47. Shatah J., Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), 313-327.
  48. Shatah J., Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), 701-710.
  49. Shatah J., Strauss W., Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173-190.
  50. Soffer A., Weinstein M.I., Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), 119-146.
  51. Soffer A., Weinstein M.I., Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), 376-390.
  52. Soffer A., Weinstein M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9-74, chao-dyn/9807003.
  53. Strauss W.A., Decay and asymptotics for u = f(u), J. Funct. Anal. 2 (1968), 409-457.
  54. Strauss W.A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162.
  55. Tao T., A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ. 4 (2007), 1-53, math.AP/0611402.
  56. Temam R., Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.
  57. Titchmarsh E., The zeros of certain integral functions, Proc. London Math. Soc. 25 (1926), 283-302.
  58. Vakhitov M.G., Kolokolov A.A., Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron. 16 (1973), 783-789.


Previous article   Next article   Contents of Volume 4 (2008)