We describe the interaction pattern in the x-y plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, (−4u t + u xxx + 6uu x ) x + 3u yy = 0. The solutions considered also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for y → ±∞, and we show that all the solutions (except the 1-soliton solution) are of resonant type, consisting of arbitrary numbers of line solitons in both asymptotics; that is, arbitrary N − incoming solitons for y → −∞ interact to form arbitrary N + outgoing solitons for y → ∞. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a web-like structure having (N − − 1)(N + − 1) holes.

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.

We demonstrate that the fission of higher-order N-solitons with a subsequent ejection of fundamental quasi-solitons creates solitonic cavities, formed by a pair of solitons with dispersive light trapped between them. As a result of multiple reflections of the trapped light from the bounding solitons which act as mirrors, they bend their trajectories and collide. In the spectral-domain, the two solitons receive blue and red wavelength shifts, respectively. The spectrum of the bouncing trapped light alters as well. This phenomenon strongly affects spectral characteristics of the generated supercontinuum. Studies of the system's parameters, which are responsible for the creation of the cavities, reveal possibilities of predicting and controlling soliton-soliton collisions induced by multiple reflections of the trapped light.

Spectrum cascade in drift wave turbulence in a magnetized plasma as well as Rossby wave turbulence in an atmospheric pressure system are studied based on a three-wave decay process derivable from the model equation applicable to both cases. The decay in the three-way interaction occurs to smaller and larger values of ‖k‖. In a region of large wavenumbers this leads to the dual cascade; the energy spectrum cascades to smaller ‖k‖ and the enstrophy spectrum to larger ‖k‖, similar to the case of two-dimensional Navier–Stokes turbulence. In a small wavenumber region a resonant three-wave decay process dominates the cascade process, and an anisotropic spectrum develops. As a consequence of the cascade, zonal flows in the direction perpendicular to the direction of inhomogeneity appear which presents a potential implication for the particle confinement in a turbulent plasma.