Related papers: Integration of differential equations by $\mathcal…
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
For systems of evolutionary partial differential equations the tau-structure is an important notion which originated from the deep relation between integrable systems and quantum field theories. We show that, under a certain non-degeneracy…
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based…
We consider overdetermined systems of difference equations for a single function $u$ which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order…
The solvability for infinite dimensional differential algebraic equations possessing a resolvent index and a Weierstra{\ss} form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which…
To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods which are based on integrable scalar nonlinear partial…
We present two integrable discretisations of a general differential-difference bicomponent Volterra system. The results are obtained by discretising directly the corresponding Hirota bilinear equations in two different ways. Multisoliton…
In the article differential-difference (semi-discrete) lattices of hyperbolic type are investigated from the integrability viewpoint. More precisely we concentrate on a method for constructing generalized symmetries. This kind integrable…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability…
The discrete autonomous/non-autonomous Toda equations and the discrete Lotka-Volterra system are important integrable discrete systems in fields such as mathematical physics, mathematical biology and statistical physics. They also have…
We consider a class of systems of difference equations defined on an elementary quadrilateral of the ${\mathbb{Z}}^2$ lattice, define their eliminable and dynamical variables, and demonstrate their use. Using the existence of infinite…
The work deals with the studies of the existence of solutions of an integro-differential equation in the situation of the difference of the standard Laplacian and the bi-Laplacian in the diffusion term. The proof of the existence of…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
The sets of the integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.
We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically…
The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…