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Asymptotic stability of solitary waves for the regularized long-wave equation. (English) Zbl 0854.35102

Summary: We show that a family of solitary waves for the regularized long-wave (RLW) equation, \[ (I- \partial^2_x) \partial_t u+ \partial_x(u+ \textstyle{{1\over 2}} u^2)= 0, \] is asymptotically stable. The large-time dynamics of a solution near a solitary wave are studied by decomposing the solution into a modulating solitary wave, with speed and phase shift that are functions of \(t\), plus a perturbation. The strategy of proof follows that used by R. L. Pego and the second author [Commun. Math. Phys. 164, No. 2, 305-349 (1994; Zbl 0805.35117)], who considered the asymptotic stability of solitary waves of Korteweg-de Vries-(KdV-)type equations. For RLW it is necessary to modify the basic ansatz to incorporate a new time scale, which must be determined by the scheme.
Different techniques are also required to analyze the spectral theory of the differential operator that arises in the linearized equation for a solitary-wave perturbation. In particular, we use a result of J. Prüss [Trans. Am. Math. Soc. 284, 847-857 (1984; Zbl 0572.47030)] to show that the linearized operator generates a semigroup with exponentially decaying norm on a certain weighted function space, and we exploit the formal convergence of RLW to KdV under a certain scaling (KdV scaling) in order to rule out the existence of nonzero eigenvalues of the linearized operator.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B25 Solitary waves for incompressible inviscid fluids
47D03 Groups and semigroups of linear operators
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