Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we consider the numerical stability of gravity-capillary wavesgenerated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) frameworkand the polynomial chaos.The stability studies are focused on the symmetric solitary wavefor the subcritical flowwith 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 randomnesswhile 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 usedfor 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.