## 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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### References:

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