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 RM . Each soliton solution defined on Gr(N, M) asymptotically consists of the N number of line-solitons for y 0 and the M −N number of line-solitons for y 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.