可精确求解与可积系统

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons which converges…

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us…

We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr\"{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization…

We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries…

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics though the time-dependent conformal mapping $z=x+iy=z(w,t)$ of the lower complex half plane of the conformal variable $w$ into the…

We perform a detailed classification of the Lie point symmetries and of the resulting similarity transformations for the Generalized Boiti-Leon-Pempinelli equations. The latter equations for a system of two nonlinear 1+2 partial…

We study the nonlinear Schr\"odinger equation on the half-line with a boundary condition that involves time derivative. This boundary condition was presented by Zambon [J. High Energ. Phys. 2014 (2014) 36]. We establish the integrability of…

We consider the dynamical stability of periodic solutions of integrable equations with 2x2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines…

We study a vector generalizations of the lattice KdV equation and one of the simplest Yamilov equations. We use algebraic properties of a certain class of matrices to derive the N-soliton solutions.

Using modified Riemann-Liouville derivative, the Exp function and Exponential rational function methods are implemented to solve the time-fractional generalized Burgers-Fisher equation (TF-GBF). The TF-GBF is transformed into a nonlinear…

An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of…

We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the…

We present exact soliton solutions of anti-self-dual Yang-Mills equations for G=GL(N) on noncommutative Euclidean spaces in four-dimension by using the Darboux transformations. Generated solutions are represented by quasideterminants of…

We analyze the dynamical behavior of the $N$-soliton train in the adiabatic approximation of the Manakov model. %perturbed by gain/loss effects and also by several types of external potentials. The evolution of Manakov $N$-soliton trains is…

This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers,…

We study discretization of Darboux integrable systems. The discretization is done by using $x$- or $y$-integrals of the considered systems. New examples of semi-discrete Darboux integrable systems are obtained.

The paper presents an approach to derive finite genus solutions to the lattice potential Kadomtsev-Petviashvili (lpKP) equation introduced by F.W. Nijhoff, et al. This equation is rederived from compatible conditions of three replicas of…

We present a novel 8-parameter integrable map in $\mathbb{R}^4$. The map is measure-preserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.

We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is…

We consider differential-difference equations that determine the continuous symmetries of discrete equations on the triangular lattice. It is shown that a certain combination of continuous flows can be represented as a scalar evolution…