One- and two-soliton solutions to a new KdV evolution equation with nonlinear and nonlocal terms for the water wave problem

Abstract

With the help of the Boussinesq perturbation expansion, a new basic equation describing the long, small-amplitude, unidirectional wave motion in shallow water with surface tension is derived to fourth order, namely a higher-order Korteweg–de Vries (KdV) equation. The procedure for deriving this equation assumes that the relation between the small parameter \(\alpha \), which measures the ratio of wave amplitude to undisturbed fluid depth, and the small parameter \(\beta \), which measures the square of the ratio of fluid depth to wave length, is taken in the form \(\beta = 0(\alpha ) = \varepsilon \), where \(\varepsilon \) is a small, dimensionless parameter which is the order of the amplitude of the motion. Hirota’s bilinear method is used to investigate one- and two-soliton solutions for this new higher-order KdV equation.