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Asymptotic stability of solitary waves

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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.

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Authors and Affiliations

  1. Department of Mathematics & Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    Robert L. Pego

  2. Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI, USA

    Michael I. Weinstein

Authors
  1. Robert L. Pego
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  2. Michael I. Weinstein
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Communicated by T. Spencer

Supported by NSF Grants DMS 9196155 and 9201869

Supported by NSF Grant DMS 9201717

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Pego, R.L., Weinstein, M.I. Asymptotic stability of solitary waves. Commun.Math. Phys. 164, 305–349 (1994). https://doi.org/10.1007/BF02101705

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  • DOI: https://doi.org/10.1007/BF02101705

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