Overtaking collision between two solitary waves

Abstract

Under the assumptions of long wavelength, small amplitude, and propagation in one direction, it is well known that the water wave equations in the lowest order of approximation with slow variation in time and space can be reduced to the Korteweg–de Vries (KdV) equation. Linear inhomogeneous equations with their coefficients, as well as the inhomogeneous terms depending on the solutions of the lower-order equations, have been obtained in the next order and the third order of approximation. An accurate numerical method is used to integrate these equations for an overtaking interaction between two solitary waves. It is found, up to the second order of approximation, that the collision is elastic, i.e., the solitary waves regain their original form without giving rise to any secondary wave. The results of second-order solution are in good agreement with the analytic solution of Sachs [SIAM J. Math. Anal. 15, 468 (1984)]. However, the numerical computation on the third-order equation shows a secondary wave train trailing behind the smaller wave after the collision.