Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension

Abstract

We derive two new two-dimensional third- and fifth-order nonlinear evolution equations that model a unidirectional wave motion in shallow water waves with surface tension. These equations are obtained through the Boussinesq perturbation expansion to one and second order, respectively, where the Taylor expansion is truncated to a finite number of terms, with respect to $$\beta = 0(\gamma ) = \varepsilon $$ , from the original system of Euler equations and surface conditions, where $$\varepsilon $$ is a small dimensionless amplitude parameter and $$\beta $$ and $$\gamma $$ measure the square of the ratio of fluid depth to wavelength, respectively, in the $$\textit{x}$$ - and $$\textit{y}$$ -direction. The one-soliton solutions of sech-squared type for these two new two-dimensional third- and fifth-order nonlinear evolution equations have been handled by the Hirota bilinear method.