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.
Similar content being viewed by others
References
Adachi, S.: A Positive solution of a nonhomogeneous elliptic equation in with G-invariant nonlinearity. Comm. PDE. 27(1&2), 1–22 (2002)
Arnol’d, V.I.: Mathematical methods of classical mechanics. Number 60 in Graduate Texts in Mathematics. New York, Springer-Verlag. Second edition, 1989
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)
Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82(4), 313–345 (1983)
Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82(4) 347–375 (1983)
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)
Bronski, J.C., Jerrard, R.L.: Soliton dynamics in a potential. Math. Res. Lett. 7(2-3), 329–342 (2000)
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)
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)
Buslaev, V.S., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. IHP. Analyse Nonlinéaire 20, 419–475 (2003)
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
Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85(4), 549–561 (1982)
Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54(9), 1110–1145 (2001)
Cuccagna, S.: Asymptotic stability of the ground states of the nonlinear Schrödinger equation. Rend. Istit. Mat. Univ. Trieste, 32(suppl. 1), 105–118 (2002)
Derks, G., van Groesen, E.: Energy propagation in dissipative systems. Part II: Centrovelocity for nonlinear wave equations. Wave Motion 15, 159–172 (1992)
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)
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)
Ginibre, J., Velo, G.: On a Class of nonlinear Schrödinger equations. I,II. J. Func. Anal. 32, 1–71 (1979)
Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equation with nonlocal interaction. Math. Z. 170(2), 109–136 (1980)
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)
Grillakis, M., Shatah, H., Strauss, W.: Stability theory of solitary waves in the presence ofsymmetry. II. J. Funct. Anal. 94(2), 308–348 (1990)
Groesen, E.S., Mainardi, F.: Energy propagation in dissipative systems. Part I: Centrovelocity for linear systems. Wave Motion 11, 201–209 (1989)
Gustafson, S., Sigal, I.M.: Dynamics of magnetic vortices. Preprint, 2003, arcXiv: math.AP/0312438
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)
Kato, T.: On Nonlinear Schrödinger Equations. Ann. IHP. Physique Théorique 46, 113–129 (1987)
Li, C., Li, Y.Y.: Nonautonomous nonlinear scalar field equations in J. Diff. Eqn. 103(2), 421–436 (1993)
Li, Y.Y.: Nonautonomous nonlinear scalar field equations. Indiana Univ. Math. J. 39(2), 283–301 (1990)
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)
McLeod, K.: Uniqueness of positive radial solutions of Am. Math. Soc. 339(2), 495–50 (1993)
McLeod, K., Serrin, J.: Uniqueness of positive radial solutions of Arch. Rational Mech. Anal. 99(2), 115–145 (1987)
Peletier, L.A., Serrin, J.: Uniqueness of positive solutions of semilinear equations in Arch. Rational Mech. Anal. 81(2), 181–197 (1983)
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)
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
Perel’man, G.S.: Preprint, 2001
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. New York: Academic Press, 1978
Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of n-soliton states of NLS. http://arxiv.org/abs/math.AP/0309114, 2003
Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133(1), 119–146 (1990)
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)
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
Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55(2), 149–162 (1977)
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)
Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Number 139 in Applied Mathematical Sciences. New York: Springer, 1999
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)
Tsai, T.-P., Yau, H.-T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002)
Tsai, T.-P., Yau, H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Comm. PDE 27(11-12), 2363–2402 (2002)
Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)
Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. XXXIX, 51–68 (1986)
Author information
Authors and Affiliations
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].
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1128-1