We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to previously known line-soliton solutions, this class also contains a large variety of new multi-soliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the taufunction, we explicitly characterize the incoming and outgoing line-solitons of this class of solutions. We illustrate these results by discussing several examples.

In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190 (2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 ( 2006)), we found a large variety of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are solitary waves which decay exponentially in (x, y)-plane except along certain rays. In this paper, we show that those solutions are classified by asymptotic information of the solution as |y| → ∞. Our study then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations. Contents1 The KPII equation and its line-soliton solutions 2 The KPII τ-function and asymptotic line-solitons

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.

The Kadomtsev-Petviashvili II (KPII) equation admits a large variety of multi-soliton solutions which exhibit both elastic as well as inelastic types of interactions. This work investigates a general class of multi-solitons which were not previously studied, and which do not in general conserve the number of line solitons after interaction. The incoming and outgoing line solitons for these solutions are explicitly characterized by analyzing the τ-function generating such solutions. A special family of N-soliton solutions is also considered in this article. These solutions are characterized by elastic soliton interactions, in the sense that amplitude and directions of the individual line solitons as y → ∞ are the same as those of the individual line solitons as y → −∞. It is shown that the solution space of these elastic N-soliton solutions can be classified into (2N − 1)!! disjoint sectors which are characterized in terms of the amplitudes and directions of the N line solitons.

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