C-Series Discrete Chains

We find analogues of the generalized two-dimensional Toda chains of the C^N and series with three discrete independent variables and give Lax pairs for these chains. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 208–221, February, 2006.     

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L₁^?1 L₂, where the linear differential operators L₁ and L₂ are chosen such that the ordinary differential equation (L₂ ?λL₁)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L₂, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L₁\(\tilde U\) = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution \(\tilde U\) of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.     

A classification algorithm for integrable two-dimensional lattices via Lie—Rinehart algebras

Theoretical and Mathematical Physics - We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean...     

Boundary value problem for the KDV equation on a half-line

The L-A pair corresponding to the boundary value problem with the conditionu| _x=0=a for the KdV equation is presented. A broad class of exact solutions to this equation is constructed and the conservation laws are discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 98–113, January, 1997.     

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.     

Initial Boundary Value Problem for the KdV Equation on a Semiaxis with Homogeneous Boundary Conditions

We consider the Korteweg–de Vries equation on the semiaxis with zero boundary conditions at x = 0 and arbitrary smooth decreasing initial data. We show that the problem can be effectively integrated by the inverse scattering transform method if the associated linear equation has no discrete spectrum. Under these assumptions, we prove the global solvability of the problem.     

Influence of water and aqueous carboxylic acids on properties of epoxide polymer materials

The possibility of biodegradation in model media is investigated. The relative changes in weight Δm/m ₀, electric permeability Δɛ/ɛ₀, tangent of dielectric losses angle Δtanδ/tanδ₀, and numeric values of physicomechanical H _R, σ_tensile, and ɛ_p, adhesive τ_shear and structural n _L parameters for epoxide polymers solidified by amine, anhydride, or catalytic agents upon exposure for a long period (up to 60 days) in water and aqueous carboxylic acids were determined. The changes in the specified parameters were found to depend on the nature of the solidifying agent. The validation of the model was performed by comparing data obtained under model conditions with those obtained during in-field tests using air and methane tanks.     

Kadomtsev–Petviashvili Equation on the Half-Plane

We study the boundary value problem for the Kadomtsev–Petviashvili equation on the half-plane y > 0 with a homogeneous condition along the boundary. We show that the problem can be efficiently solved using the dressing method. We present explicit solutions for particular cases of the boundary value problem.     

Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Kortewegde Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov-Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra A ₁ ⁽¹⁾ . We find new examples of integrable systems of equations on a square lattice.     

An initial-boundary value problem on the half-line for the MKdV equation

The initial-boundary value problem on the half-line for the modified Korteweg-de Vries equation with zero boundary conditions and arbitrary rapidly decaying initial conditions is embedded in the scheme of the inverse scattering method. The corresponding inverse scattering problem is reduced to the Riemann problem on a system of rays in the complex plane. Supported by the Russian Foundation for Basic Research under grant 98-01-00576. Mathematical Institute, Russian Academy of Sciences, Ufa; e-mail: ihabib@imat.rb.ru. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 65–75, January–March, 2000. Translated by I. T. Habibullin