Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation

Solutions of the non-linear hyperbolic equations describing quasi-transverse waves in composite elastic media are investigated within the framework of a previously proposed model, which takes into account small dissipative and dispersion processes. It is well known for this model that if a solution of the problem of the decay of an arbitrary discontinuity is constructed using Riemann waves and discontinuities having a structure, the solution turns out to be non-unique. In order to study the problem of non-uniqueness, solutions of non-self-similar problems are constructed numerically within the framework of the proposed model with initial data in the form of a “smooth” step. With time passing the solutions acquire a self-similar asymptotic form, corresponding to a certain solution of the problem of the decay of an arbitrary discontinuity. It is shown that, by changing the method of smoothing the step, one can construct any of the self-similar asymptotic forms, as was done previously in Ref. [Chugainova AP. The asymptotic behaviour of non-linear waves in elastic media with dispersion and dissipation. Teor Mat Fiz 2006;147(2):240–56] for media with terms of opposite sign, responsible for the non-linearity, although the set of admissible discontinuities and the structure of the solutions of the problems in these cases turn out to be different.     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

A note on the dispersion of Love waves in layered monoclinic elastic media

The dispersion equation for Love waves in a monoclinic elastic layer of uniform thickness overlying a monoclinic elastic half-space is derived by applying the traction-free boundary condition at the surface and continuity conditions at the interface. The dispersion curves showing the effect of anisotropy on the calculated phase velocity are presented. The special cases of orthotropic and transversely isotropic media are also considered. It is shown that the well-known dispersion equation for Love waves in an isotropic layer overlying an isotropic half-space follows as a particular case.     

Generation of soliton pairs in nonlinear media with weak dissipation

Weak interaction processes are studied for two KdV solitons at low viscosity. A model is suggested for describing the interaction in terms of slowly varying parameters of the exact two-soliton solution to the KdV equation under perturbation. It is shown that both the inverse scattering problem method and the Witham method lead to the same system of reduced equations. The resulting solutions are in good agreement with numerical calculation data. The main effect is the formation of a bound quasistationary soliton pair and its subsequent dissipation. It turns out that although both the process of soliton merger and the process of dissipation are due to low viscosity, the former is substantially faster. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 254–271, February, 1997.     

Controlling nonlinear waves in excitable media

A new feedback control method is proposed to control the spatio-temporal dynamics in excitable media. Applying suitable external forcing to the system’s slow variable, successful suppression and control of propagating pulses as well as spiral waves can be obtained. The proposed controller is composed by an observer to infer uncertain terms such as diffusive transport and kinetic rates, and an inverse-dynamics feedback function. Numerical simulations shown the effectiveness of the proposed feedback control approach.     

Autoresonant asymptotics in an oscillating system with weak dissipation

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 102–111, July, 2009.     

Theory of one-dimensional waves in nonlinear dissipative media

A unified approach is made to the construction of mathematical models describing discontinuous processes, and their analogs in the region of continuous solutions are presented; this makes it possible to distinguish between idealized and actual discontinuities and to classify media on the basis of model equations. The media considered are characterized by physical and geometric nonlinearity and by dissipation due to either viscosity or to the coupling of strain and temperature fields.Institute of Cybernetics, Academy of Sciences of the Estonian SSR, Tallin. Translated from Mekhanika Polimerov, No. 1, pp. 41–48, January–February, 1976.     

New results for guided waves in heterogeneous elastic media

We prove the existence of guided waves propagating with a velocity strictly larger than the S (shear) wave velocity at infinity in the case of unbounded elastic media invariant under translation in one space direction and asymptotically homogeneous at infinity. These waves correspond to the existence of eigenvalues embedded in the essential spectrum of the self-adjoint elastic propagator.     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.     

Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media

We analyze family of solutions to multidimensional scalar conservation law, with flux depending on the time and space explicitly, regularized with vanishing diffusion and dispersion terms. Under a condition on the balance between diffusion and dispersion parameters, we prove that the family of solutions is precompact in L¹_loc{L^1_{rm loc}}. Our proof is based on the methodology developed in Sazhenkov (Sibirsk Math Zh 47(2):431–454, 2006), which is in turn based on Panov’s extension (Panov and Yu in Mat Sb 185(2):87–106, 1994) of Tartar’s H-measures (Tartar in Proc R Soc Edinb Sect A 115(3–4):193–230, 1990), or Gerard’s micro-local defect measures (Gerard Commun Partial Differ Equ 16(11):1761–1794, 1991). This is new approach for the diffusion–dispersion limit problems. Previous results were restricted to scalar conservation laws with flux depending only on the state variable.