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Asymptotic stability of solitary waves. (English) Zbl 0805.35117

Summary: We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation \(\partial_ tu + u \partial_ xu + \partial^ 3_ xu = 0\), is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, \[ \partial_ tu + \partial_ xf(u) + \partial^ 3_ xu = 0. \] In particular, we study the case where \(f(u) = u^{p + 1}/(p + 1)\), \(p = 1,2,3\) (and \(3<p<4\), for \(u>0\), with \(f \in C^ 4)\). The same asymptotic stability result for KdV is also proved for the case \(p=2\) (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 of \(p\) between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameter \(p\) increases beyond the critical value \(p=4\).)
The solution is decomposed into a modulating solitary wave, with time- varying speed \(c(t)\) and phase \(\gamma(t)\) (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. As \(p \to 4^ -\), the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B35 Stability in context of PDEs
35Q51 Soliton equations
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