Abstract
We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation
is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations,
In particular, we study the case wheref(u)=u p+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C 4). 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.
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References
[AS] Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981
[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)
[Be] Benjamin, T.B.: The stability of solitary waves. Proc. Roy. Soc. Lond.A328, 153–183 (1972)
[Ber] Berryman, J.G.: Stability of solitary waves in shallow water. Phys. Fluids19, 771–777 (1976)
[Bo] Bona, J.L.: On the stability of solitary waves. Proc. Roy. Soc. Lond.A344, 363–374 (1975)
[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)
[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)
[BSo] Bona, J.L., Soyeur, A.: On the stability of solitary wave solutions of model equations for long waves. J. Nonlin. Science
[BSS1] Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves. Proc. Roy. Soc. Lond.A411, 395–412 (1987)
[CL] Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955
[C] Coppel, W.A.: Stability and asymptotic behavior of differential equations. Boston: D.C. Heath and Co., 1965
[CS] Crandall, M.G., Souganidis, P.E.: Convergence of difference approximations of quasilinear evolution equations. Nonl. Anal. TMA10, 425–445 (1986)
[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)
[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)
[DJ] Drazin, P.G., Johnson, R.S.: Solitons: An introduction. Cambridge: Cambridge University Press, 1989
[E] Evans, J.W.: Nerve axon equations, IV: The stable and unstable impulse. Indiana Univ. Math. J.24, 1169–1190 (1975)
[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)
[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)
[GKZ] Gesztesy, F., Karwowski, K., Zhao, Z.: New types of soliton solutions. Bull. Am. Math. Soc.27, 266–272 (1992)
[GT] Ginibre, J., Tsutsumi, Y.: Uniqueness for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal.20, 1388–1425 (1989)
[H] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math.840, New York: Springer, 1981
[JK] Jeffrey, A., Kakutani, T.: Weakly nonlinear dispersive waves: A discussion centered about the Korteweg-de Vries equation. SIAM Review14, 582–643 (1972)
[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, 1975
[K2] Kato, T.: Perturbation Theory for Linear Operators. 2nd ed., New York: Springer, 1976
[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, 1983
[Ke] Keener, J.P.: Principles of Applied Mathematics, Redwood City: Addison-Wesley, 1988
[KM] Keener, J.P., McLaughlin, D.W.: Solitons under perturbations. Phys. Rev.A 16, 777–790 (1977)
[KA] Kodama, Y., Ablowitz, M.J.: Perturbations of solitons and solitary waves. Stud. in Appl. Math.64, 225–245 (1981)
[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)
[L] Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.21, 467–490 (1968)
[LP] Lax, P.D., Phillips, R.S.: Scattering Theory. New York: Academic Press, 1967
[LS] Laedke, E.W., Spatschek, K.H.: Stability theorem for KdV type equations. J. Plasma Phys.32, 263–272 (1984)
[M] Miura, R.M.: Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation. J. Math. Phys.9, 1202–1204 (1968)
[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. 127–179
[Ni] Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute Lecture Notes, 1974
[P] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci.44, New York: Springer, 1983
[PW1] Pego, R.L., Weinstein, M.I.: On asymptotic stability of solitary waves. Phys. Lett. A162, 263–268 (1992)
[PW2] Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Phil. Trans. Roy. Soc. Lond.A340, 47–94 (1992)
[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, 1993
[RS3] Reed, M., Simon, B.: Methods of Modern Mathematical Physics III: Scattering Theory. New York: Academic Press, 1979
[RS4] Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press, 1978
[Sa] Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. in Math.22, 312–355 (1976)
[Sc] Schuur, P.C.: Asymptotic Analysis of Soliton Problems. Lect. Notes in Math.1232, Berlin New York: Springer 1986
[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, 1988
[SW2] Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering theory for nonintegrable equations. Commun. Math. Phys.133, 119–146 (1990)
[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)
[T] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Oxford: Oxford University Press, 1946
[V] Vainberg, B.: Asymptotic Methods in Equations of Mathematical Physics. New York: Gordon and Breach, 1989
[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, 1986
[W2] Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 472–491 (1985)
[W3] Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math.39, 51–68 (1986)
[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)
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Communicated by T. Spencer
Supported by NSF Grants DMS 9196155 and 9201869
Supported by NSF Grant DMS 9201717
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Pego, R.L., Weinstein, M.I. Asymptotic stability of solitary waves. Commun.Math. Phys. 164, 305–349 (1994). https://doi.org/10.1007/BF02101705
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DOI: https://doi.org/10.1007/BF02101705