Numerical stability analysis of steady solutions for the forced KdV equation based on the polynomial chaos expansion

Two-dimensional gravity-capillary waves can be modeled by the forced Korteweg-de Vries (fKdV) equation in subcritical flows when the Bond number is greater than one third. Four steady symmetric depression wave solutions and two elevation wave solutions for the fKdV equation have been found and time evolutions of their magnitude or spatial perturbations have been observed. We approach the fKdV equation as a stochastic equation by modeling the perturbation as a random variable and examine the stabilities of the steady solutions based on the polynomial chaos expansion framework. Polynomial chaos also provides surfaces, which encompass random fluctuations of unstable waves. The effects of several parameters on the stabilities and the surfaces have been also considered.