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Solitary Wave Dynamics in an External Potential

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Abstract

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton’s equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

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References

  1. Adachi, S.: A Positive solution of a nonhomogeneous elliptic equation in with G-invariant nonlinearity. Comm. PDE. 27(1&2), 1–22 (2002)

    Google Scholar 

  2. Arnol’d, V.I.: Mathematical methods of classical mechanics. Number 60 in Graduate Texts in Mathematics. New York, Springer-Verlag. Second edition, 1989

  3. Berestycki, H., Gallouet, T., Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297(5), 307–310 (1983)

    MATH  Google Scholar 

  4. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82(4), 313–345 (1983)

    MATH  Google Scholar 

  5. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82(4) 347–375 (1983)

    Google Scholar 

  6. Berestycki, H., Lions, P.-L., Peletier, L.A.: An ODE approach to the existence of positive solutions for semilinear problems in Indiana Univ. Math. J. 30(1) 141–157 (1981)

    Google Scholar 

  7. Bronski, J.C., Jerrard, R.L.: Soliton dynamics in a potential. Math. Res. Lett. 7(2-3), 329–342 (2000)

    Google Scholar 

  8. Buslaev, V.S., Perel’man, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz 4(6), 63–102 (1992)

    Google Scholar 

  9. Buslaev, V.S., Perel’man, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. Am. Math. Soc. Transl. Ser. 2(164), 74–98 (1995)

    Google Scholar 

  10. Buslaev, V.S., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. IHP. Analyse Nonlinéaire 20, 419–475 (2003)

    Article  MATH  Google Scholar 

  11. Cazenave, T.: An introduction to nonlinear Schrödinger equations. Number 26 in Textos de Métodos Matemáticos. Rio de Janeiro RJ: Instituto de Matematica - UFRJ, Third edition, 1996

  12. Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85(4), 549–561 (1982)

    MATH  Google Scholar 

  13. Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54(9), 1110–1145 (2001)

    Article  MATH  Google Scholar 

  14. Cuccagna, S.: Asymptotic stability of the ground states of the nonlinear Schrödinger equation. Rend. Istit. Mat. Univ. Trieste, 32(suppl. 1), 105–118 (2002)

  15. Derks, G., van Groesen, E.: Energy propagation in dissipative systems. Part II: Centrovelocity for nonlinear wave equations. Wave Motion 15, 159–172 (1992)

    Article  MATH  Google Scholar 

  16. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On a classical limit of quantum theory and the non-linear Hartree equation. Geom. Funct. Anal. Special Volume, Part I, 57–78 (2000)

  17. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Comm. Math. Phys. 225(2), 223–274 (2002)

    Article  Google Scholar 

  18. Ginibre, J., Velo, G.: On a Class of nonlinear Schrödinger equations. I,II. J. Func. Anal. 32, 1–71 (1979)

    Google Scholar 

  19. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equation with nonlocal interaction. Math. Z. 170(2), 109–136 (1980)

    MATH  Google Scholar 

  20. Grillakis, M., Shatah, H., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987)

    MATH  Google Scholar 

  21. Grillakis, M., Shatah, H., Strauss, W.: Stability theory of solitary waves in the presence ofsymmetry. II. J. Funct. Anal. 94(2), 308–348 (1990)

    MATH  Google Scholar 

  22. Groesen, E.S., Mainardi, F.: Energy propagation in dissipative systems. Part I: Centrovelocity for linear systems. Wave Motion 11, 201–209 (1989)

    MATH  Google Scholar 

  23. Gustafson, S., Sigal, I.M.: Dynamics of magnetic vortices. Preprint, 2003, arcXiv: math.AP/0312438

  24. Jones, C.K.R.T., Küpper, T.: On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 17(4), 803–835 (1986)

    MATH  Google Scholar 

  25. Kato, T.: On Nonlinear Schrödinger Equations. Ann. IHP. Physique Théorique 46, 113–129 (1987)

    MATH  Google Scholar 

  26. Li, C., Li, Y.Y.: Nonautonomous nonlinear scalar field equations in J. Diff. Eqn. 103(2), 421–436 (1993)

    Google Scholar 

  27. Li, Y.Y.: Nonautonomous nonlinear scalar field equations. Indiana Univ. Math. J. 39(2), 283–301 (1990)

    MATH  Google Scholar 

  28. Lions, P.-L.: On positive solutions of semilinear elliptic equations in unbounded domains. Nonlinear diffusion equations and their equilibrium states II (Berkeley CA 1986), Math. Sci. Res. Inst. Publ. 13, 85–122 (1988)

  29. McLeod, K.: Uniqueness of positive radial solutions of Am. Math. Soc. 339(2), 495–50 (1993)

    Google Scholar 

  30. McLeod, K., Serrin, J.: Uniqueness of positive radial solutions of Arch. Rational Mech. Anal. 99(2), 115–145 (1987)

    Google Scholar 

  31. Peletier, L.A., Serrin, J.: Uniqueness of positive solutions of semilinear equations in Arch. Rational Mech. Anal. 81(2), 181–197 (1983)

    Google Scholar 

  32. Pelinovsky, D.E., Afanasjev, V.V., Kivshar, Y.S.: Nonlinear theory of oscillating decaying and collapsing solitons in the general nonlinear Schrödinger equation. Phys. Rev. E 53(2), 1940–53 (1996)

    Article  Google Scholar 

  33. Pelinovsky, D.E., Grimshaw, R.H.J.: Asymptotic methods in soliton stability theory. In: L. Debnath and S.R. Choudhury (eds.), Nonlinear instability analysis, Vol. 12, Comput. Mech. Southampton, 1997, pp. 245–312

  34. Perel’man, G.S.: Preprint, 2001

  35. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. New York: Academic Press, 1978

  36. Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of n-soliton states of NLS. http://arxiv.org/abs/math.AP/0309114, 2003

  37. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133(1), 119–146 (1990)

    MATH  Google Scholar 

  38. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data. J. Differ. Eqs. 98(2), 376–390 (1992)

    MATH  Google Scholar 

  39. Soffer, A., Weinstein, M.I.: Selection of the ground state for nonlinear Schrödinger equations. Preprint 2001, revised 2003, http://arxiv.org/abs/nlin.PS/0308020, 2003

  40. Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55(2), 149–162 (1977)

    MATH  Google Scholar 

  41. Stuart, D.M.A.: Modulation approach to stability of non-topological solitions in semilinear wave equations. J. Math. Pures Appl. 80(1), 51–83 (2001)

    Article  MATH  Google Scholar 

  42. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Number 139 in Applied Mathematical Sciences. New York: Springer, 1999

  43. Tsai, T.-P., Yau, H.-T. Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Comm. Pure Appl. Math. 55(2), 153–216 (2002)

    Article  MATH  Google Scholar 

  44. Tsai, T.-P., Yau, H.-T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tsai, T.-P., Yau, H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Comm. PDE 27(11-12), 2363–2402 (2002)

    Google Scholar 

  46. Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)

    MATH  Google Scholar 

  47. Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. XXXIX, 51–68 (1986)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute für Theoretische Physik, ETH Hönggerberg, Zürich, Switzerland

    J. Fröhlich

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    S. Gustafson

  3. The Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada

    B.L.G. Jonsson

  4. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    B.L.G. Jonsson & I.M. Sigal

  5. The Alfvénlaboratory, Royal Institute of Technology, Stockholm, Sweden

    B.L.G. Jonsson

  6. Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA

    I.M. Sigal

Authors
  1. J. Fröhlich
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  2. S. Gustafson
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  3. B.L.G. Jonsson
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  4. I.M. Sigal
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Additional information

Communicated by M. Aizenman

Supported by NSERC grant 22R80976

The support of Wenner-Gren Foundation is gratefully acknowledged

Supported partially by NSERC under NA7601 and by NSF under DMS-0400526

Acknowledgement. B.L.G.J. and I.M.S. are grateful to J. Colliander for useful discussions and remarks and to ETH-Zürich for hospitality during their work on this paper. J.F. thanks T.-P. Tsai and H.-T. Yau for very useful discussions and correspondence which led to the results in [16,17].

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Fröhlich, J., Gustafson, S., Jonsson, B. et al. Solitary Wave Dynamics in an External Potential. Commun. Math. Phys. 250, 613–642 (2004). https://doi.org/10.1007/s00220-004-1128-1

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  • DOI: https://doi.org/10.1007/s00220-004-1128-1

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