Two-dimensional interaction of weakly nonlinear solitary waves in shallow water: the Benney-Luke equation and the KP equation

Kenichi Maruno (University of Texas Pan American)

Friday 5th July, 2013 15:30-16:30 Maths 416


Weakly nonlinear shallow water waves are described by the Benney-Luke (BL) equation.  Assuming weak two-dimensionality, the Kadomtsev-Petviashvili (KP) equation is derived. The KP equation is known as one of integrable systems, and its solutions are written in explicit forms. Various interesting solutions, such as P-type, T-type, O-type, (3,1,2,4)-type, of the KP equation have been found by Kodama and his collaborators.

Recently, Kodama and Williams discovered a way to solve inverse problems of KP line solitons by
using the theory of total positivity for the Grassmannian.  However, it is not so clear whether these solutions of the KP equation and the theory to solve inverse problems is applicable to real shallow water wave phenomena since weak two-dimensionality was assumed in the derivation of the KP equation. The BL equation is a better model of shallow water waves although the BL equation is not integrable. Thus we need to clarify the difference of solutions of the BL and the KP equations. We propose a method to obtain an approximate 2 soliton solution for the BL equation. This method is based on the Hirota bilinear method and the reductive perturbation method. Using this 2-soliton solution, we compute critical angles of transitions between different solitary wave interactions. The accuracy of these critical angles are confirmed by using direct numerical simulations of the BL equation. We also discuss the application to Mach reflection in shallow water waves.

This is joint work with Yuji Kodama (Ohio State University), Hidekazu Tsuji (Kyushu University), and Baofeng Feng (University of Texas - Pan American).

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