Soliton solutions of the KP equation and application to shallow water waves
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The main purpose of this paper is to give a survey of recent development on a classification of soliton solutions of the KP equation. The paper is self-contained, and we give a complete proof for the theorems needed for the classification. The classification is based on the Schubert decomposition of the real Grassmann manifold, Gr$(N,M)$, the set of $N$-dimensional subspaces in $\mathbb{R}^M$. Each soliton solution defined on Gr$(N,M)$ asymptotically consists of the $N$ number of line-solitons for $y\gg 0$ and the $M-N$ number of line-solitons for $y\ll 0$. In particular, we give the detailed description of those soliton solutions associated with Gr$(2,4)$, which play a fundamental role of multi-soliton solutions. We then consider a physical application of some of those solutions related to the Mach reflection discussed by J. Miles in 1977.
49 pages, 22 figures, Submitted for the conference proceedings "Nonlinearwave 2008" at Beijing, June 2008
49 pages, 22 figures, Submitted for the conference proceedings "Nonlinearwave 2008" at Beijing, June 2008