ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation $$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$ is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, $$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 .$$ In particular, we study the case wheref(u)=u p+1/(p+1),p=1, 2, 3 (and 3〈p〈4, foru〉0, withf∈C 4). The same asymptotic stability result for KdV is also proved for the casep=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 ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(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. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02101705
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