Asymptotics for the Korteweg-de Vries-Burgers equation

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

Abstract

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut + uux ? uxx + u xxx = 0, x ε R, t > 0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u 0 ε Hs (R) ∩ L 1 (R), where s > - 1/2, then there exists a unique solution u (t, x) ε C ((0,∞) ;H (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t) = t-1/2 fM (( ·)t-1/2) + o(t -1/2) as t → ∞, where f M is the self-similar solution for the Burgers equation. Moreover if xu 0 (x) L 1 (R) , then the asymptotics are true u(t) = t-1/2 f M (( ·)t-1/2) + O(t-1/2-γ) where γ ε (0,1/2).

Original languageEnglish
Pages (from-to)1441-1456
Number of pages16
JournalActa Mathematica Sinica, English Series
Volume22
Issue number5
DOIs
Publication statusPublished - 2006 Sep
Externally publishedYes

Keywords

  • Asymptotics for large time
  • Korteweg-de Vries-Burgers equation
  • Large initial data

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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