Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type

E. Tzirtzilakis, V. Marinakis, C. Apokis, T. Bountis

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

In this work we study second and third order approximations of water wave equations of the Korteweg-de Vries (KdV) type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitude-velocity dependence are identical to the sech2 formula of the one-soliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme, in parameter regions where soliton-like behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves (trailing the solitary wave) and may still be regarded as stable, provided these radiation waves do not exceed a numerical stability threshold. Instability occurs at high enough wave speeds, when these oscillations exceed the stability threshold already at the outset, and manifests itself as a sudden increase of these oscillations followed by a blowup of the wave after relatively short time intervals.

Original languageEnglish
Pages (from-to)6151-6165
Number of pages15
JournalJournal of Mathematical Physics
Volume43
Issue number12
DOIs
Publication statusPublished - Dec 1 2002
Externally publishedYes

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Soliton-like Solutions
Higher order equation
Solitary Waves
wave equations
Wave equation
solitary waves
Oscillation
Solitons
Exceed
Radiation
Pseudospectral Method
Fourier Method
Wave Speed
Solitary Wave Solution
Water Waves
Numerical Stability
Approximation
Soliton Solution
Finite Difference Scheme
General Solution

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type. / Tzirtzilakis, E.; Marinakis, V.; Apokis, C.; Bountis, T.

In: Journal of Mathematical Physics, Vol. 43, No. 12, 01.12.2002, p. 6151-6165.

Research output: Contribution to journalArticle

Tzirtzilakis, E, Marinakis, V, Apokis, C & Bountis, T 2002, 'Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type', Journal of Mathematical Physics, vol. 43, no. 12, pp. 6151-6165. https://doi.org/10.1063/1.1514387
Tzirtzilakis, E. ; Marinakis, V. ; Apokis, C. ; Bountis, T. / Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type. In: Journal of Mathematical Physics. 2002 ; Vol. 43, No. 12. pp. 6151-6165.
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