Related papers: Towards the classification of integrable different…
We develop a theory of integrable dispersive deformations of 2+1 dimensional Hamiltonian systems of hydrodynamic type following the scheme proposed by Dubrovin and his collaborators in 1+1 dimensions. Our results show that the…
An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type u_t+A(u)u_x+B(u)u_y=0 is developed. It is proved that the existence of special solutions known as `double waves' is…
We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method…
A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
After extending the Clarkson-Kruskal's direct similarity reduction ansatz to a more general form, one may obtain various new types of reduction equations. Especially, some lower dimensional turbulence systems or chaotic systems may be…
This paper is devoted to constructing and studying exactly solvable dynamical systems in discrete time obtained from some algebraic operations on matrices, to reductions of such systems leading to classical field theory models in…
We construct a family of integrable hydrodynamic type systems with three independent and n>1 dependent variables in terms of solutions of linear system of PDEs with rational coefficients. We choose the existence of a pseudopotential as a…
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…
We have proposed, in our previous papers, a method to characterize integrable discrete soliton equations. In this paper we generalize the method further and obtain a $q$-difference Toda equation, from which we can derive various…
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
This paper is devoted to the study of the singularly perturbed second order partial integro-differential equations. The estimation of the solutions of Cauchy problem is obtained.
Nonlinear integrable models with two spatial and one temporal variables: Kadomtsev-Petviashvili equation and two-dimensional Toda lattice are investigated on the subject of correct formulation for boundary problem that can be solved within…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
In the paper we study some numerical solutions to Volterra equations which interpolate heat and wave equations. We present a scheme for construction of approximate numerical solutions for one and two spatial dimensions. Some solutions to…
In this paper we shall study differential equations in the complex domain. The method of indeterminate coefficients and the majorant method lead to a proof of the existence and uniqueness of meromorphic solution of differential equations.…
We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter $\ep$. We apply the prolongation method, the…
Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes…