可精确求解与可积系统
We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and…
We study the class of cubic Hamiltonian vector fields whose associated Kahan-Hirota-Kimura (KHK) maps preserve the original Hamiltonian function. Our analysis focuses on these fields in $\mathbb{R}^2$ and $\mathbb{R}^4$, extending to a…
In this paper we discuss some remarkable properties of the autonomous system of 2 first-order Ordinary Differential Equations (ODEs), which equates the derivatives $\dot{x}_n(t)$ ($n = 1, 2$) of the 2 dependent variables $x_n(t)$ to the…
We address the long-time asymptotics of the solution to the Cauchy problem of ccSP (coupled complex short pulse) equation on the line for decaying initial data that can support solitons. The ccSP system describes ultra-short pulse…
An effective method for constructing explicit solutions to the Davey--Stewartson type integrable equations is discussed based on the use of a dressing chain. The application of the method is exemplified by the equation DS I, for which a new…
Using the methods of quantisation ideals, we construct a family of quantisations corresponding to Case alpha in Sergeev's classification of solutions to the tetrahedron equation. This solution describes transformations between special…
The delay Lotka-Volterra and delay Toda lattice equations are delay-differential extensions of the well-known soliton equations, the Lotka-Volterra and Toda lattice equations, respectively. This paper investigates integrable properties of…
We describe the Ozawa solution to the Davey--Stewartson II equation from the point of view of surface theory by presenting a soliton deformation of surfaces which is ruled by the Ozawa solution. The Ozawa solution blows up at certain moment…
The real and imaginary parts of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field $\mathbb{C}$ give rise to the focusing gauged MKdV (FGMKdV) equations. As a generalization of Euler's elastica whose…
The negative order KdV (nKdV) and the modified KdV (nmKdV) equations have two different formulations based on different hierarchy operators. Both equations can be written in terms of a nonlinear differential equation for a field $u(x,t)$…
We study the integrability of the quantized six-vertex model with four parameters on a torus. It is a three-dimensional integrable lattice model in which a layer transfer matrix, depending on two spectral parameters associated with the…
We propose the Lax-pair representation for an integrable semi-discretization (discretization of the spatial variable) of the massive Thirring model in non-characteristic (in between light-cone and laboratory) coordinates and present its…
A negative symmetry is a nonlocal symmetry of special type. In this paper, we introduce a method for constructing negative symmetries from consistent triplets of differential and differential-difference equations. Moreover, we study the…
We formulate the inverse spectral theory for a non-self-adjoint one-dimensional Dirac operator associated periodic potentials via a Riemann-Hilbert problem approach. We use the resulting formalism to solve the initial value problem for the…
We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant…
In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations (in the sense having 3-soliton solutions) of the…
For the finite (non-periodic) systems obtained from a lattice introduced by Ferapontov and independently by Shabat and Yamilov, we present a quadrature-free general solution and a recurrent formula for the characteristic integrals. The…
We present a method for constructing hierarchies of solutions to $n$-simplex equations by variating the spectral parameter in their Lax representation. We use this method to derive new solutions to the set-theoretical 2- and 3-simplex…
Hex systems were recently introduced [A. P. Kels. Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces. 2022. arXiv:2205.02720 [math-ph]] as systems of equations defined on…
Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3\psi/dx^3+Q\,d\psi/dx+P\psi…