Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension

Abstract

This paper deals with the water waves problem for uneven bottom under the influence of surface tension. We consider here an asymptotic limit for the Green–Naghdi equations in KdV scale, that is the Boussinesq system. The derivation of the KdV equation with uneven bottom under the influence of surface tension has been established. Indeed, this derivation is obtained in a formal way by using the Whitham technique, then the analytic solution to this equation has been obtained in case of flat bottom. However, in case of uneven bottom an $$H^s$$ -consistent solution has been obtained. Also, an $$H^s$$ -consistent solution for the Boussinesq system has been established, taking into consideration the influence of surface tension and uneven bottom. Finally, we confirmed the obtained theoretical results of this paper numerically, by devoting the last section to make a numerical validation. Moreover, analytic solutions to the KdV $$\sigma $$ and Boussinesq system were established in case of several bottom parametrization including the linear one.