Alexey V. Porubov

Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer

Abstract An exact general periodical solution in terms of the Weierstrass elliptic function ℘ is obtained to the nonlinear dissipative ODE governing free surface travelling waves on a viscous convecting liquid layer. This solution may describe, in particular, the bounded periodical wave only in the presence of a nonlinear dissipative term in the equation studied.

Long non-linear strain waves in layered elastic half-space

Elastic strain wave propagation in a thin non-linearly elastic layer superimposed on non-linearly elastic half-space is studied. Two layer-half-space contact models are considered. It is found that the Benjamin-Ono equation can be derived for description of longitudinal non-linear strain waves, when the contact between the layer and the half-space is provided only by means of the normal stresses and displacements. When the full contact problem is considered the more complicated integro-differential equation is derived. It is found that long non-linear periodical and solitary strain waves as well as envelope waves may exist in the first case, while only envelope wave solutions are found to the full contact problem. Linear wave analysis shows that the Korteveg-de Vries equation, often usable, is unlikely to be an adequate model for longitudinal surface strain waves. Application of the results obtained to experiments devoted to superconductivity threshold control in thin metal films as well as to generation of acoustic solitons in layered half-space is discussed.

Reflection of a Longitudinal Strain Soliton from the End Face of a Nonlinearly Elastic Rod

Reflection of a solitary longitudinal strain wave (soliton) from the end face of a nonlinearly elastic rod is investigated theoretically and experimentally. It is shown that the wave reflected from the free end of the rod has a reversed amplitude sign, which results in dispersion of the wave. If the end of the rod is fixed, the reflected wave retains its polarity and properties of the incident solitary wave and propagates back to the input end.

Holographic interferometry of strain waves as a tool for nondestructive testing

The paper presents a new approach to detect impurities, inhomogeneities and anisotropy in condensed matter based on laser generation and holographic observation of nonlinear waves in solids, both transparent and opaque. This approach uses the two phenomena recently discovered by us. First is generation of so-called Poisson waves in a medium surrounding the solid when the strain wave propagates inside it. Second, our studies have proved the physical possibility to generate and observe waves of a new type -- longitudinal strain solitons in nonlinearly elastic waveguides. The soliton detection and recording allows to introduce a new one-pulse technology in nondestructive testing, to determine intrinsic physical properties of nonlinearly elastic materials.

On the localization of 2D nonlinear internal waves in a two-layer fluid

A 2D generalized Gardner equation is used to describe 2D nonlinear internal waves in a two-layer fluid. Unlike the previous model based on the Kadomtsev-Petviashvili equation, the model considered here allows for the instability of a plane internal solitary wave. Such a possibility causes the wave to be localized in any direction. Relationships between the thicknesses and densities of the layers under the instability conditions are obtained.

Nonlocal Approach to Square Lattice Dynamics

The algorithm is developed to model two-dimensional dynamic processes in a nonlocal square lattice on the basis of the shift operators. The governing discrete equations are obtained for local and nonlocal models. Their dispersion analysis reveals important differences in the dispersion curve and in the sign of the group velocity caused by nonlocality. The continuum limit allows to examine possible auxetic behavior of the material described by the nonlocal discrete model.

On the role of cubic nonlinearity in localization of longitudinal strain waves

New governing equations with combined quadratic and cubic nonlinearities are obtained to account for nonlinear strain waves in an elastic rod and in a plate. It is shown that strain solitary wave solutions of these equations arise as a result of balance between quadratic nonlinearity and dispersion and exists even in the absence of cubic nonlinearity. However, the amplitude, the width and the velocity of the wave are affected by the cubic nonlinearity causing, in particular, a narrowing of the longitudinal solitary wave. This allows to agree better with experiments on strain solitary wave generation in the rod and in the plate.

Holographic studies of strain soliton evolution in nonlinearly elastic solid waveguides

Results are presented from theory and experiments carried out to excite and observe a localized nonlinear longitudinal strain wave (soliton) in different waveguides. The wave parameters are calculated. It is shown that such a wave conserves its profile during propagation in an elastic waveguide.

Laser generation and optical observation of strain solitons in nonlinearly elastic solid waveguides

The possibility of generation and observation of a nonlinear solitary strain wave (soliton) in different solid waveguides is of special interest in connection with undeniable potential applications in solid state physics and mechanics. The present paper deals with experimental results on the studies of soliton generation and evolution in a relatively long waveguide as well as in a waveguide of varying cross section (inhomogeneous waveguide).

Application of holographic interferometry for strain soliton observation in nonlinear elastic solid waveguides

The possibility of generation and observation of a nonlinear solitary strain wave (soliton) in a solid waveguide is of special interest in connection with undeniable potential applications in solid state physics and mechanics. The present paper deals with new results on the soliton evolution in a relatively long waveguide as well as in a waveguide of varying cross sections.