### Abstract

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitarywave- like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. First it allows us to identify the stable solution of the stochastic governing equation. Secondly it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.

Original language | English |
---|---|

Pages (from-to) | 833-847 |

Number of pages | 15 |

Journal | Advances in Applied Mathematics and Mechanics |

Volume | 4 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2012 Dec 1 |

### Fingerprint

### Keywords

- Forced Korteweg-de Vries equation
- Polynomial chaos
- Solitary waves
- Stability

### ASJC Scopus subject areas

- Applied Mathematics
- Mechanical Engineering

### Cite this

**Stability of symmetric solitarywave solutions of a forced korteweg-de vries equation and the polynomial Chaos.** / Kim, Hongjoong; Moon, Kyoung Sook.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics and Mechanics*, vol. 4, no. 6, pp. 833-847. https://doi.org/10.4208/aamm.12-12S012

}

TY - JOUR

T1 - Stability of symmetric solitarywave solutions of a forced korteweg-de vries equation and the polynomial Chaos

AU - Kim, Hongjoong

AU - Moon, Kyoung Sook

PY - 2012/12/1

Y1 - 2012/12/1

N2 - In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitarywave- like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. First it allows us to identify the stable solution of the stochastic governing equation. Secondly it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.

AB - In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitarywave- like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. First it allows us to identify the stable solution of the stochastic governing equation. Secondly it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.

KW - Forced Korteweg-de Vries equation

KW - Polynomial chaos

KW - Solitary waves

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=84873312500&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873312500&partnerID=8YFLogxK

U2 - 10.4208/aamm.12-12S012

DO - 10.4208/aamm.12-12S012

M3 - Article

VL - 4

SP - 833

EP - 847

JO - Advances in Applied Mathematics and Mechanics

JF - Advances in Applied Mathematics and Mechanics

SN - 2070-0733

IS - 6

ER -