Related papers: KP solitons and the Schottky uniformization
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to…
We show that the $\tau$-functions of the regular KP solitons from the totally nonnegative Grassmannians can be expressed by the Riemann theta functions on singular curves. We explicitly write the parameters in the Riemann theta function in…
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to…
One constructs the parity-time symmetric solitons in the complex KP Equation using the totally non-negative Grassmannian. We obtain that every element in the totally non-negative orthogonal Grassmannian corresponds to a parity-time…
We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the…
Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x,…
The KP equation is a nonlinear dispersive wave equation which provides an excellent model for resonant interactions of shallow-water waves. It is well known that regular soliton solutions of the KP equation may be constructed from points in…
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.…
It is known that soliton solutions of the KP-hierarchy corresponds to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves.…
In order to generalize a program by Agostini and others producing new KP solutions, we construct families of quasi-periodic KP solutions which are derived from degenerating Riemann surfaces associated with tropical curves having nontrivial…
We consider the Kadomtsev-Petviashvili II (KP) model placed in $\mathbb R_t \times \mathbb R_{x,y}^2$, in the case of smooth data that are not necessarily in a Sobolev space. In this paper, the subclass of smooth solutions we study is of…
The BKP equation is obtained from the reduction of B type in the KP hierarchy under the orthogonal type transformation group for the KP equation. The skew Schur Q functions can be used to construct the Tau functions of solitons in the BKP…
We give a classification of the $regular$ soliton solutions of the KP hierarchy, referred to as the $KP solitons$, under the Gel'fand-Dickey $\ell$-reductions in terms of the permutation of the symmetric group. As an example, we show that…
It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(\Omega,\Gamma,P:\Omega \to S)$, where $\Gamma$ is a Schottky…
Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x,…
The solitons solution of BKP equation can be constructed by the Pfaffian structure. Then one investigates the real line solitons structure of BKP equation using the totally non-negative Grassmannian. Especially, the N-soliton solution is…
We survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background…
Using the Wronskian representation of $\tau$-function, one can investigate the resonant structure of kink-soliton and line-soliton of the modified KP equation. It is found that the resonant structure of the the soliton graph is obtained by…
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The classification is based on the far-field patterns of the solutions which consist of a finite…
We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. Those solutions also satisfy the finite Toda lattice hierarchy.…