Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Invariant difference schemes for the Ermakov system

We construct invariant difference schemes for the parametric system of Ermakov equations. By using a difference analog of the Noether theorem, we write out the first three difference integrals of the system. The obtained schemes are integrable exactly to the same extent to which the original differential system is integrable.     

First integrals of difference equations which do not possess a variational formulation

The paper presents a new method for finding first integrals of difference equations which do not possess Lagrangians, nor Hamiltonians. In this paper we consider ordinary differential and difference equations. As an example we solve a third order nonlinear ordinary differential equation and its invariant discretization using three first integrals obtained by means of this method.     

Identification of parameters of a plane elliptic crack in an isotropic linearly elastic body from the results of a single uniaxial tension test

The method earlier developed by one of the authors for identifying ellipsoidal defects is numerically tested for the applicability to the problem of identification of a degenerate ellipsoidal defect, i.e., an elliptic crack. The method is based on the reciprocity functional and the assumption that the displacements are measured in a uniaxial tension test of an isotropic linearly elastic body. Calculations show that the earlier developed method is also efficient for identification of an elliptic crack and its parameters (the center coordinates, the normal to the crack plane, and the directions and lengths of the semiaxes) can be determined with high accuracy. Some examples where the crack has a non-elliptic shape are also considered. It is discovered that, in many cases, the ellipsoids that were constructed by formulas reconstructing the ellipsoidal crack from the data on the external boundary of the body that correspond to a nonelliptic crack, approximate the actual defect with sufficient accuracy. The method stability was investigated with respect to noise in the initial data.     

Peculiarities of the surface and guided waves propagation in heterogeneous composites with nematic coatings

Using the effective low frequency model of nematic medium we study the phase speeds and attenuations of the quasi-Rayleigh and Love waves in composite made of elastic core with the peripheral coating, as well as the fundamental S0 and A0 Lamb modes in heterogeneous plates. The analytical and numerical analysis revealed two types of unusual resonance effects disturbing wave propagation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some classes of two-dimensional vortex flows of an ideal fluid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 109–117, January–February, 1989.     

Construction of exact solutions of the Boussinesq equation

The known model of nonlinear dispersive waves, which was proposed by Boussinesq in the second half of the nineteenth century, is considered. Solutions of the Boussinesq equation, which are expressed via elementary functions and describe wave packets, their interaction between each other and with solitons, and some other structures are obtained. To construct these solutions, Hirota's bilinear representation and differential relations specified by ordinary differential equations with constant coefficients are used. Computing Center, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 74–78, May–June, 1998.