The decay of the continuous spectrum for solutions of the Korteweg-deVries equation

Mark J. Ablowitz, Alan C. Newell

Research output: Contribution to journalArticle

56 Scopus citations

Abstract

The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation ut + uux + uxxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x ≫ t1/3, (ii) x = 0(t1/3), (iii) x ≪ - t1/3. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t1/2 when - x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(- x/t)1/4t1/2.

Original languageEnglish (US)
Pages (from-to)1277-1284
Number of pages8
JournalJournal of Mathematical Physics
Volume14
Issue number9
DOIs
StatePublished - Jan 1 1973
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint Dive into the research topics of 'The decay of the continuous spectrum for solutions of the Korteweg-deVries equation'. Together they form a unique fingerprint.

  • Cite this