Soliton solutions of the KP equation and application to shallow water waves

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.