Re-visiting the head-on collision problem between two solitary waves in shallow water

Abstract

Upon discovering the wrongness of the statement “although this term does not cause any secularity for this order it will cause secularity at higher order expansion, therefore, that term must vanish” by Su and Mirie [4], in the present work, we studied the head-on collision of two solitary waves propagating in shallow water by introducing a set of stretched coordinates in which the trajectory functions are of order of ϵ2, where ϵ is the smallness parameter measuring non-linearity. Expanding the field variables and trajectory functions into power series in ϵ, we obtained a set of differential equations governing various terms in the perturbation expansion. By solving them under non-secularity condition we obtained the evolution equations and also the expressions for phase functions. By seeking a progressive wave solution to these evolution equations we have determined the speed correction terms and the phase shifts. As opposed to the result of Su and Mirie [4] and similar works, our calculations show that the phase shifts depend on both amplitudes of the colliding waves.