On the asymptotic integration of a linear system of differential equations with a small parameter in the coefficients of some derivatives

Using a transformation matrix, we reduce a system of differential equations with a small parameter in the coefficients of some derivatives and a turning point to an integrable system of equations. We also study properties of the transformation matrix.     

Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients and prove a theorem on the estimation of its precision. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 122–132, January, 2007.     

The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension

The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.     

Distortion of a matrix structure and the cluster formation in MnxZnyFezO4 ferrite single crystals

The evolution of the hard and soft modes in the crystal structure of Mn_xZn_yFe_zO₄ manganese-zinc spinel ferrites is investigated theoretically and experimentally (x-ray structure analysis and magnetic measurements). It is shown that tetragonal and rhombohedral distortions of the cubic structure and fluctuations of the soft mode are accompanied by the formation of clusters in the form of quasi-two-dimensional fragments (50–200 and 400–1200 Å), which involve families of γ-Mn₂O₃, γ-Mn₃O₄, and α-Fe₂O₃ oxide planes inherently bound to the spinel matrix.     

On the Existence of Periodic Solutions for Certain Classes of Systems of Differential Equations with Random Pulse Influence

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.     

Structure and frustration in liquid crystalline polyacrylates II. Thin-film properties

We report on the thin-film behaviour of a polyacrylate with phenyl benzoate mesogenic side groups and a narrow polydispersity. Depending on the degree of polymerisation, these polymers show a nematic, a smectic-A _d, a re-entrant nematic and a C phase with a two-dimensional monoclinic lattice. X-ray reflectivity and atomic-force microscopy have been used to characterize the structure and surface morphology. The system exhibits two stable side-chain packing configurations with incommensurate spacings that can be both stabilized at a free surface. Thin films in the nematic phase show a structural dewetting induced by the growth of surface domains of the C phase. Additionally, surface-induced ripples with a nanoscale lateral period form at the air-film interface. We attribute these patterns to a coupling between the local liquid crystalline ordering of the mesogenic side groups and the surface curvature energy. Received 28 February 2001 and Received in final form 6 August 2001     

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.