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On the stability of solitary-wave solutions of model equations for long waves. (English) Zbl 0809.35095

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.

MSC:

35Q51 Soliton equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
35S10 Initial value problems for PDEs with pseudodifferential operators
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