Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos
Year: 2012
Author: Hongjoong Kim, Kyoung-Sook Moon
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 6 : pp. 833–847
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 solitary-wave-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. Firstly, 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.12-12S12
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 6 : pp. 833–847
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Stability solitary waves polynomial chaos forced Korteweg-de Vries equation.