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

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.