### 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 sech^{2} 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 language | English |
---|---|

Pages (from-to) | 6151-6165 |

Number of pages | 15 |

Journal | Journal of Mathematical Physics |

Volume | 43 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 1 2002 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Mathematical Physics*,

*43*(12), 6151-6165. https://doi.org/10.1063/1.1514387

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

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 43, no. 12, pp. 6151-6165. https://doi.org/10.1063/1.1514387

}

TY - JOUR

T1 - Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type

AU - Tzirtzilakis, E.

AU - Marinakis, V.

AU - Apokis, C.

AU - Bountis, T.

PY - 2002/12/1

Y1 - 2002/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0036930496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036930496&partnerID=8YFLogxK

U2 - 10.1063/1.1514387

DO - 10.1063/1.1514387

M3 - Article

VL - 43

SP - 6151

EP - 6165

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

ER -