Asymptotic method for the water wave equation.

By an asymptotic procedure, the complete water wave equation is shown to have a reflected wave after overtaking collisions of two solitary waves [C.-H. Su and Q. S. Zou, Phys. Rev. A 35, 4738 (1987)]. This is a result of a nonlocal behavior of the solution. Using the asymptotic procedure to a model equation of water waves, which is shown to be solvable by the inverse scattering method [D.J. Kaup, Progr. Theor. Phys. 54, 396 (1975)], we show that the solution to the model equation does not have such a nonlocal behavior. Hence the asymptotic method is consistent with the inverse scattering method. It provides a complementary way to check complete integrability for some systems in addition to the inverse scattering method and the Painlev\'e analysis, etc.