Related papers: KP Solitons from Tropical Limits
We study here the Zakharov-Kuznetsov equation in dimension $2$ and $3$ and the modified Zakharov-Kuznetsov equation in dimension $2$. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of…
We introduce the notion of abelian solutions of the 2D Toda lattice equations and the bilinear discrete Hirota equation and show that all of them are algebro-geometric.
We construct quasi-periodic solutions of the universal hierarchy which includes the multi-component KP and Toda hierarchies and show how they fit into the bilinear formalism. The tau-function is expressed in terms of the Riemann…
In this letter we examine the two-component Hirota (TH) equations which describes the pulse propagation in a coupled fiber with higher-order dispersion and self-steepening. As the TH equations is a complete integrable system, which admits a…
Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over…
Blending Painlev\'e property with singularity confinement for a general arbitrary order Sawada-Kotera differential-difference equation, we find a proliferation of ``tau-functions'' (coming from strictly confined patterns). However only one…
We identify the Kontsevich-Penner matrix integral, for finite size $n$, with the isomonodromic tau function of a $3\times 3$ rational connection on the Riemann sphere with $n$ Fuchsian singularities placed in correspondence with the…
We discuss some properties of the soliton equations of the type, partial derivative u/partial derivative t = S [u, (u) over bar], where S is a nonlinear operator differential in x, and present the additivity theorems of the class of the…
We announce a detailed investigation of limits of N-soliton solutions of the Korteweg-deVries (KdV) equation as $N$ tends to infinity. Our main results provide new classes of KdV-solutions including in particular new types of soliton-like…
In this paper, we are concerned with various soliton solutions to the coupled Hirota equation, as well as to the Sasa-Satsuma equation which can be viewed as one reduction case of the coupled Hirota equation. First, we derive bright-bright,…
This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the…
Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta…
A large family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is…
It is shown that, by letting wavenumbers and frequencies complex in Hirota's bilinear method, new classes of exact solutions of soliton equations can be obtained systematically. They include not only singular or N-homoclinic solutions but…
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the…
We show that when KP (Kadomtsev-Petviashvili) $\tau$ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete…
The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev-Petviashvili (KP) equation. This is derived algebraically from a Fredholm…
Tropical R is the birational map that intertwines products of geometric crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and C^{(1)}_n are…
Using the second flow - the Derivative Reaction-Diffusion system, and the third one of the dissipative SL(2,R) Kaup-Newell hierarchy, we show that the product of two functions, satisfying those systems is a solution of the modified…
The novel generalized perturbation (n, M)-fold Darboux transformations (DTs) are reported for the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation and its extension by using the Taylor expansion of the Darboux matrix. The generalized…