KP solitons in shallow water
Abstract
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the KadomtsevPetviashvili (KP) equation. The KP equation describes weakly dispersive and small amplitude wave propagation in a quasitwodimensional framework. Recently, a large variety of exact soliton solutions of the KP equation has been found and classified. These solutions are localized along certain lines in a twodimensional plane and decay exponentially everywhere else, and are called linesolitons. The classification is based on the farfield patterns of the solutions which consist of a finite number of linesolitons. Each soliton solution is then defined by a point of the totally nonnegative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain classes of initial waves asymptotically approach to these exact solutions of the KP equation. We discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extraordinary fourfold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included in the quasitwodimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 October 2010
 DOI:
 10.1088/17518113/43/43/434004
 arXiv:
 arXiv:1004.4607
 Bibcode:
 2010JPhA...43Q4004K
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Physics  Fluid Dynamics
 EPrint:
 50 pages, 25 figures