Line-soliton solutions of the KP equation

Abstract

A review of recent developments in the study and classification of the line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation is provided. Such solution u(x, y,t) is defined by a point of the totally non-negative Grassmannian Gr(N, M), and for fixed t, decays exponentially except along N distinct directions for y 0 and M − N distinct directions for y 0. These solutions can be constructed algebraically as u = 2(ln τ) xx where the function τ is a wronskian of N independent solutions of f y = f xx and f t = f xxx . The KP equation admits a large variety of such multi-soliton solutions in contrast to its one-dimensional counterpart -the Kortweg de-Vries (KdV) equation. It is shown how these solutions can be classified using combinatorial methods.should admit solutions that provide good approximations for two soliton interaction with smaller angle. Thus, it seems strange that one does not have a reasonable solution in the parameter regimes where the KP equation is supposed to give a better approximation. Miles also found that at the critical angle the two line-solitons of the O-type solution interact resonantly, and a third wave (soliton) is created to make a "Y-shape" solution. Indeed, it turns out that such Y-shape resonant solutions are exact solutions of the KP equation (see also [25]). Miles then used the Y-shape solution to describe the Mach reflection observed in shallow water waves reflected by a rigid wall. However the recent papers [7,19] shows that the Mach reflection should be described by other resonant solution of the KP equation (described also in the present paper).After the discovery of the resonant phenomena in the KP equation, several numerical and experimental studies were performed to investigate resonant interactions in other physical two dimensional equations such as the ion-acoustic and shallow water wave equations (see for examples [15,16,26,11,22]). However, after these activities, no significant progress has been made in the study of the solution space or physical applications of the KP equation. It would appear that the general perception was that there were not many new and significant results left to be uncovered in the soliton solutions of the KP theory. During the past 5 years, we have studied the classification problem of the soliton solutions of the KP equation, and our investigation has revealed a large variety of soliton solutions which were totally overlooked in the past. The purpose of this paper is to give a survey of our recent works [4, 18, 3, 5, 6, 7, 8] on a classification theory of soliton solutions of the Kadomtsev-Petviashvili (KP) equation. Most of the materials in this paper except for the discussion of Darboux transformations in Section 3 were presented by one of the authors (YK) as a tutorial at the international workshop "Nonlinear and Modern Mathematical Physics" held in Beijing during July 15-21, 2009. The organization of the paper is as follows. In Section 2, we provide some background material by introducing the wronskian form of the KP τ-function. Th...

KP line solitons and Tamari lattices

“…At t + the original slower soliton swallows the virtual one and mutates to x 12,24 x 14 , which is a new manifestation of the original faster soliton. A generalization of this solution, now with n parallel line solitons 14 , is given by τ = (e 1 − (−1) n e 2n ) ∧ (e 2 − (−1) n−1 e 2n−1 ) ∧ · · · ∧ (e n−1 − e n+2 ) ∧ (e n + e n+1 ) , where p 1 < p 2 < · · · < p 2n and p 1 + p 2n = p 2 + p 2n−1 = · · · = p n−1 + p n+2 = p n + p n+1 . Moreover, by taking the wedge product of two such functions, we can generate grid-like structures.…”

“…More comprehensive studies of the structure of the rather complex networks emerging in this way have been undertaken quite recently [6][7][8][9][10][11][12][13][14][15][16][17][18] (see also the review [19] and the references cited therein). Whereas in these works a classification in terms of the asymptotic behavior at large negative and positive times, and large (positive or negative) values of the coordinate transverse to the main propagation direction, has been addressed, in the present work we proceed toward an understanding of the full evolution.…”

Remarks on combinatorial aspects of the KP equation

“…Since we are following the main lines of enquiry in [10,12,11,29,28], the remainder of this paper concerns Hirota integrability of the KP-II Equation, and in particular, combinatorics of certain Wronskian solutions. We will restrict our attention to this family of solutions to (1) and refer the reader to [21] for a careful discussion of other determinantal methods used in the study of bilinearizable PDE (e.g., Grammians, Pfaffians, Casoratians, etc.…”

“…We shall refer to the associated functions τ (and consequently u in (1)) as soliton solutions of the equation. For a complete discussion, see [12,26,21,34], for instance. Now that we have established formulas for the p-derivatives of the exponentials φ j (p, x), we wish to use them as "generating seeds" in the construction of Wronskian solutions to (3).…”

“…The main purpose of this communication is to survey a recent successful approach for analyzing various soliton solutions of the KP equation using combinatorics appearing in the body of work [10,11,27,29,28,12,26]; the general method is to generate a family of soliton solutions to the KP equation with a certain combinatorial structure, and then analyze the KP solutions produced in light of this underlying structure. This gives a bijection between combinatorial objects (e.g.…”