Summary
After a review of the existing state of affairs, an improvement is made in the stability theory for solitary-wave solutions of evolution equations of Korteweg-de Vries-type modelling the propagation of small-amplitude long waves. It is shown that the bulk of the solution emerging from initial data that is a small perturbation of an exact solitary wave travels at a speed close to that of the unperturbed solitary wave. This not unexpected result lends credibility to the presumption that the solution emanating from a perturbed solitary wave consists mainly of a nearby solitary wave. The result makes use of the existing stability theory together with certain small refinements, coupled with a new expression for the speed of propagation of the disturbance. The idea behind our result is also shown to be effective in the context of one-dimensional regularized long-wave equations and multidimensional nonlinear Schrödinger equations.
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References
Albert, J. P. 1992. Positivity properties and stability of solitary-wave solutions of model equations for long waves.Commun. Partial Differential Equations 17, 1–22.
Albert, J. P. and Bona, J. L. 1991. Total positivity and the stability of internal waves in fluids of finite depth.IMA J. Appl. Math. 46, 1–19.
Albert, J. P., Bona, J. L., and Henry, D. 1987. Sufficient conditions for stability of solitary-wave solutions of model equations for long waves.Phys. D 24, 343–366.
Benjamin, T. B. 1972. The stability of solitary waves.Proc. Roy. Soc. London A 328, 153–183.
Benjamin, T. B., Bona, J. L., and Bose, D. K. 1990. Solitary-wave solutions of nonlinear problems.Philos. Trans. Roy. Soc. London A 331, 195–244.
Bennett, D. P., Brown, R. W., Stansfield, S. E., Stroughair, J. D., and Bona, J. L. 1983. The stability of internal solitary waves in stratified fluids.Math. Proc. Cambridge Philos. Soc. 94, 351–379.
Berestycki, H., Lions, P.-L., and Peletier, L. A. 1981. An ODE approach to the existence of positive solutions for semilinear problems in ℝN.Indiana Univ. Math. J. 30, 141–157.
Bona, J. L. 1975. On the stability theory of solitary waves.Proc. Roy. Soc. London A 349, 363–374.
Bona, J. L., Dougalis, V. A., and Karakashian, O. A. 1986. Fully discrete Galerkin methods for the Korteweg-de Vries equation.J. Comp. Math. Appl. 12A, 859–884.
Bona, J. L., Dougalis, V. A., Karakashian, O. A., and McKinney, W. 1991. Fully-discrete methods with grid refinement for the generalized Korteweg-de Vries equation. InViscous Profiles and Numerical Methods for Shock Waves (M. Shearer, ed.), pp. 1–11. SIAM, Philadelphia.
Bona, J. L., Dougalis, V. A., Karakashian, O. A., and McKinney, W. 1994. Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation. To appear.
Bona, J. L., Dougalis, V. A., Karakashian, O. A., and McKinney, W. 1993. The effect of dissipation on solutions of the generalized Korteweg-de Vries equation. Unpublished.
Bona, J. L. and Sachs, R. 1988. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation.Commun. Math. Phys. 118, 15–29.
Bona, J. L., Souganidis, P. E., and Strauss, W. A. 1987. Stability and instability of solitary waves of KdV type.Proc. Roy. Soc. London A 411, 395–412.
Cazenave, T. 1989.An Introduction to Nonlinear Schrödinger Equations. Textos de Métodos Matemáticas22. UFRJ, Rio de Janiero.
Cazenave, T. and Lions, P.-L. 1982. Orbital stability of standing waves for some nonlinear Schrödinger equations.Commun. Math. Phys. 85, 549–561.
Eckhaus, W. and Schuur, P. 1983. The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions.Math. Methods Appl. Sci. 5, 97–116.
Grillakis, M., Shatah, J., and Strauss, W. A. 1987. Stability of solitary waves in the presence of symmetry, I.J. Funct. Anal. 74, 160–197.
Landau, L. D. and Lifshitz, E. M. 1958.Quantum Mechanics—Nonrelativistic Theory. Addison-Wesley, Reading, MA.
Maddocks, J. and Sachs, R. 1993. On the stability of the KdV multi-solitons.Commun. Pure Appl. Math. 46 (6), 867–901.
Pego, R. and Weinstein, M. 1992. On asymptotic stability of solitary waves.Phys. Lett. A 162, 263–68.
Pego, R. and Weinstein, M. 1993. Eigenvalues and instability of solitary waves.Philos. Trans. Roy. Soc. London A 340, 47–94.
Scott, A. C., Chu, F. Y., and McLaughlin, D. W. 1973. The soliton: a new concept in applied science.Proc. IEEE 61 (10), 1443–1483.
Strauss, W. A. 1977. Existence of solitary waves in higher dimensions.Commun. Math. Phys. 55, 149–162.
Strauss, W. A. and Souganidis, P. E., 1990. Instability of a class of dispersive solitary waves.Proc. Roy. Soc. Edinburgh 114A, 195–212.
Warchall, H. 1986. A wave operator for the scattering of solitary waves in multiple spatial dimensions. InNonlinear Systems of Partial Differential Equations in Applied Mathematics (B. Nicolaenko, D. D. Holm, and J. M. Hyman, eds.).Lectures on Applied Math.23. American Math. Soc., Providence.
Weinstein, M. 1986. Lyapunov stability of ground states of nonlinear dispersive evolution equations.Commun. Pure Appl. Math. 39, 51–68.
Weinstein, M. 1987. Existence and dynamic stability of solitary-wave solutions of equations arising in long wave propagation.Commun. Partial Differential Equations 12, 1133–1173.
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Communicated by Thanasis Fokas
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Bona, J.L., Soyeur, A. On the stability of solitary-wave solutions of model equations for long waves. J Nonlinear Sci 4, 449–470 (1994). https://doi.org/10.1007/BF02430641
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DOI: https://doi.org/10.1007/BF02430641
Key words
- solitary waves
- stability
- nonlinear dispersive wave equations
- model equations for long waves
- Korteweg-de Vries-type equations
- regularized long-wave equations
- nonlinear Schrödinger equations