Communications in Mathematical Physics

, Volume 164, Issue 2, pp 305–349 | Cite as

Asymptotic stability of solitary waves

  • Robert L. Pego
  • Michael I. Weinstein
Article
Received:

Abstract

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation
$$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$
is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations,
$$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 .$$
In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C4). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.

Keywords

Radiation Neural Network Complex System Nonlinear Dynamics Evolution Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [AS] Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981Google Scholar
  2. [AGJ] Alexander, J., Gardner, R., Jones, C.K.R.T.: A topological invariant arising in the stability analysis of traveling waves. J. Reine Angew. Math.410, 167–212 (1990)Google Scholar
  3. [Be] Benjamin, T.B.: The stability of solitary waves. Proc. Roy. Soc. Lond.A328, 153–183 (1972)Google Scholar
  4. [Ber] Berryman, J.G.: Stability of solitary waves in shallow water. Phys. Fluids19, 771–777 (1976)Google Scholar
  5. [Bo] Bona, J.L.: On the stability of solitary waves. Proc. Roy. Soc. Lond.A344, 363–374 (1975)Google Scholar
  6. [BSc] Bona, J.L., Scott, R.: Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces. Duke Math J.43, 87–99 (1976)Google Scholar
  7. [BSm] Bona, J.L., Smith, R.: The initial value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. Lond.A278, 555–601 (1975)Google Scholar
  8. [BSo] Bona, J.L., Soyeur, A.: On the stability of solitary wave solutions of model equations for long waves. J. Nonlin. ScienceGoogle Scholar
  9. [BSS1] Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves. Proc. Roy. Soc. Lond.A411, 395–412 (1987)Google Scholar
  10. [CL] Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955Google Scholar
  11. [C] Coppel, W.A.: Stability and asymptotic behavior of differential equations. Boston: D.C. Heath and Co., 1965Google Scholar
  12. [CS] Crandall, M.G., Souganidis, P.E.: Convergence of difference approximations of quasilinear evolution equations. Nonl. Anal. TMA10, 425–445 (1986)Google Scholar
  13. [CH1] Crawford, J.D., Hislop, P.D.: Application of the method of spectral deformation to the Vlasov Poisson system. Ann. Phys.189, 265–317 (1989)Google Scholar
  14. [CH2] Crawford, J.D., Hislop, P.D.: Application of the method of spectral deformation to the Vlasov Poisson system II. Mathematical results. J. Math. Phys.30, 2819–2837 (1989)Google Scholar
  15. [DJ] Drazin, P.G., Johnson, R.S.: Solitons: An introduction. Cambridge: Cambridge University Press, 1989Google Scholar
  16. [E] Evans, J.W.: Nerve axon equations, IV: The stable and unstable impulse. Indiana Univ. Math. J.24, 1169–1190 (1975)Google Scholar
  17. [GGKM1] Gardner, C.S., Greene, J.M., Kruskal M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett.19, 1095–1097 (1967)Google Scholar
  18. [GGKM2] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Commun. Pure Appl. Math.27, 97–133 (1974)Google Scholar
  19. [GKZ] Gesztesy, F., Karwowski, K., Zhao, Z.: New types of soliton solutions. Bull. Am. Math. Soc.27, 266–272 (1992)Google Scholar
  20. [GT] Ginibre, J., Tsutsumi, Y.: Uniqueness for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal.20, 1388–1425 (1989)Google Scholar
  21. [H] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math.840, New York: Springer, 1981Google Scholar
  22. [JK] Jeffrey, A., Kakutani, T.: Weakly nonlinear dispersive waves: A discussion centered about the Korteweg-de Vries equation. SIAM Review14, 582–643 (1972)Google Scholar
  23. [K1] Kato, T.: Quasilinear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations, A. Dold and B. Eckmann, (eds.), Lect. Notes in Math.448, New York: Springer, 1975Google Scholar
  24. [K2] Kato, T.: Perturbation Theory for Linear Operators. 2nd ed., New York: Springer, 1976Google Scholar
  25. [K3] Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Stud. in Appl. Math. Suppl. Ser.8, New York: Academic Press, 1983Google Scholar
  26. [Ke] Keener, J.P.: Principles of Applied Mathematics, Redwood City: Addison-Wesley, 1988Google Scholar
  27. [KM] Keener, J.P., McLaughlin, D.W.: Solitons under perturbations. Phys. Rev.A 16, 777–790 (1977)Google Scholar
  28. [KA] Kodama, Y., Ablowitz, M.J.: Perturbations of solitons and solitary waves. Stud. in Appl. Math.64, 225–245 (1981)Google Scholar
  29. [KPV] Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J.A.M.S.4, 323–347 (1991)Google Scholar
  30. [L] Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.21, 467–490 (1968)Google Scholar
  31. [LP] Lax, P.D., Phillips, R.S.: Scattering Theory. New York: Academic Press, 1967Google Scholar
  32. [LS] Laedke, E.W., Spatschek, K.H.: Stability theorem for KdV type equations. J. Plasma Phys.32, 263–272 (1984)Google Scholar
  33. [M] Miura, R.M.: Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation. J. Math. Phys.9, 1202–1204 (1968)Google Scholar
  34. [Ne] Newell, A.C.: Near-integrable systems, nonlinear tunneling and solitons in slowly changing media. In: Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform. F. Calogero, (ed.), London: Pitman, 1978, pp. 127–179Google Scholar
  35. [Ni] Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute Lecture Notes, 1974Google Scholar
  36. [P] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci.44, New York: Springer, 1983Google Scholar
  37. [PW1] Pego, R.L., Weinstein, M.I.: On asymptotic stability of solitary waves. Phys. Lett. A162, 263–268 (1992)Google Scholar
  38. [PW2] Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Phil. Trans. Roy. Soc. Lond.A340, 47–94 (1992)Google Scholar
  39. [PW3] Pego, R.L., Weinstein, M.I.: Evans' function, Melnikov's integral and solitary wave instabilities. In: Differential Equations with Applications to Mathematical Physics. (eds.) W.F. Ames, J.V. Herod, E.M. Harrell, Orlando: Academic Press, 1993Google Scholar
  40. [RS3] Reed, M., Simon, B.: Methods of Modern Mathematical Physics III: Scattering Theory. New York: Academic Press, 1979Google Scholar
  41. [RS4] Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press, 1978Google Scholar
  42. [Sa] Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. in Math.22, 312–355 (1976)Google Scholar
  43. [Sc] Schuur, P.C.: Asymptotic Analysis of Soliton Problems. Lect. Notes in Math.1232, Berlin New York: Springer 1986Google Scholar
  44. [SW1] Soffer, A., Weinstein M.I.: Multichannel nonlinear scattering theory for nonintegrable equations. In: Integrable Systems and Applications. Balabane, M., Lochak, P. and Sulem, C. (eds.), Springer Lect. Notes in Physics,342, 1988Google Scholar
  45. [SW2] Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering theory for nonintegrable equations. Commun. Math. Phys.133, 119–146 (1990)Google Scholar
  46. [SW3] Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering theory for nonintegrable equations II: The case of anisotropic potentials and data. J. Diff. Eq.98, 376–390 (1992)Google Scholar
  47. [T] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Oxford: Oxford University Press, 1946Google Scholar
  48. [V] Vainberg, B.: Asymptotic Methods in Equations of Mathematical Physics. New York: Gordon and Breach, 1989Google Scholar
  49. [W1] Weinstein, M.I.: On the solitary wave of the generalized Korteweg-de Vries equation. In: Proc. Santa Fe Conference on Nonlinear PDE, July 1984, Lectures in Appl. Math.23, 1986Google Scholar
  50. [W2] Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 472–491 (1985)Google Scholar
  51. [W3] Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math.39, 51–68 (1986)Google Scholar
  52. [ZK] Zabusky, N., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett.15, 240–243 (1965)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Robert L. Pego
    • 1
  • Michael I. Weinstein
    • 2
  1. 1.Department of Mathematics & Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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