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 developments on a classification of soliton solutions of the Kadomtsev-Petviashvili equation. The paper is self-contained, and we give complete proofs of theorems needed for the classification. The classification is based on the totally nonnegative cells in the Schubert decomposition of the real Grassmann manifold, Gr(N, M), the set of N-dimensional subspaces in R-M. 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 detailed description of the soliton solutions associated with Gr(2, 4), which play a fundamental role in the study of multisoliton solutions. We then consider a physical application of some of those solutions related to the Mach reflection discussed by J. Miles in 1977.