On head-on collision between two solitary waves in shallow water: the use of the extended PLK method

In the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}, introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.