Related papers: Bilinear approach to Kuperschmidt super-KdV type e…
This paper presents a complete description of the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation. By constructing an approximate solution of the problem, we prove that at the main order, the two…
We show that a number of nonlinear equations including symmetric as well as asymmetric $\phi^4$, modified Korteweg de Vries (MKdV), mixed KdV-MKdV, nonlinear Schr\"odinger (NLS), quadratic-cubic NLS as well as higher order neutral scalar…
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…
We consider the Hirota equation (the discrete analog of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogs of…
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by…
We obtain novel periodic as well as hyperbolic solutions of an Ablowitz-Musslimani variant of the coupled nonlocal, nonlinear Schr\"odinger equation (NLS) as well as a coupled nonlocal modified Korteweg-de Vries (mKdV) equation which can be…
Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in $1+1$ dimensions. There are three different topological soliton solutions which be labelled according to their topological…
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM…
We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite dimensional extension of the well-known…
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…
Assuming that there exist at least two fermionic parameters, the classical N= 1 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are…
Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by means of the fermionic representation for the KdV field is considered. It is shown that in both cases the corresponding Hamiltonians are given as sums of two well…
Multiphonon processes in a model quantum dot (QD) containing two electronic states and several optical phonon modes are considered taking into account both intra- and inter-level terms. The Hamiltonian is exactly diagonalized including a…
We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved…
We derive exact analytical solutions describing multi-soliton complexes and their interactions on top of a multi-component background in media with self-focusing or self-defocusing Kerr-like nonlinearities. These results are illustrated by…
The Casimir energy for the classically stable configurations of the topological solitons in 2D quantum antiferromagnets is studied by performing the path-integral over quantum fluctuations. The magnon fluctuation around the solitons…
Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota…
A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it…
A class of "elliptic soliton" solutions of the Kadomtsev-Petviashvili hierarchy, which includes a determinantal solution of Li and Zhang, is described in terms of pseudo-differential operator formulation. In our approach, the Li-Zhang…
We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial…