可精确求解与可积系统

In this paper, we take advantage of the bilinearization reduction method to consider the local and nonlocal reduction of a discrete Ablowitz-Kaup-Newell-Segur equation. Exact solutions in double Casoratian form to the reduced nonlocal…

Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie…

The present preprint is dedicated to a nonlinear evolution equation that generalizes the classical Heisenberg ferromagnet equation in certain way. That generalization is completely integrable and has a linear bundle Lax pair in pole gauge…

In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial…

In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract…

We explore the phenomena of absorption/emission of solitons by an integrable boundary for the nonlinear Schr\"odinger equation on the half-line. This is based on the investigation of time-dependent reflection matrices which satisfy the…

We consider a degeneration of the $q$-matrix sixth Painlev\'e system. As a result, we obtain a system of non-linear $q$-difference equations, which describes a deformation of a certain non-Fuchsian linear $q$-difference system. We define…

In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with…

In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous…

We consider the problem of reflectionless propagation of PT-symmetric solitons described by the nonlocal nonlinear Schroedinger equation on a line in the framework of the concept of transparent boundary conditions for evolution equations.…

We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV, Harry Dym, coupled Harry Dym,…

The multi-soliton solution of the sine-Gordon equation in the presence of elliptic-function background is derived by the inverse scattering method. The key tool in our formulation is the Lax pair written by $4\times4$ matrix differential…

We find integrals of motion for the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. Our method is based on the…

In this article we study St\"{a}ckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable St\"{a}ckel systems of Benenti type, related with different…

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction…

We formulate and study an integrable model of Nonlinear Schr\"odinger (NLS)-type through its Lax representation, where one of the Lax operators is quadratic and the other has a rational dependence on the spectral parameter. We discuss the…

We consider discrete sine-Gordon equation on branched domains. The latter is modeled in terms of the metric graphs with discrete bonds having the form of the branched 1D chains. Exact analytical solutions of the problem are obtained for…

We obtain B\"acklund transformations and integrable time discretization of the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with…

We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. We derive the Hirota bilinear forms of these equations and obtain soliton solutions. We find all possible local and nonlocal reductions…

Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that…