On the dissipation due to wave ringing in nonelliptic elastic materials

Summary Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.     

Static Vortex Solutions in a Planar Particle System Obeying a Generalized Exclusion Principle

We consider a planar particle system obeying a generalized Pauli exclusion principle. In the mean field approximation, this system is described by a Schrödinger equation we recently introduced, containing a complex nonlinearity. The particle number, the total energy, and the angular momentum are conserved in such a system. We consider vortexlike stationary solutions of the form and write the differential equation for the vortex shape. We find an analytic solution of this equation and obtain a closed expression for the vortex profile. We investigate some mean properties and, in particular, calculate the energy spectrum and angular momentum of the vortex.     

Nonlinear transparency regimes for three-component acoustic pulses in a system of electron and nuclear spins

We study acoustic solitons consisting of one longitudinal and two transverse components and propagating in the direction perpendicular to an external magnetic field in a crystal containing paramagnetic impurities of electron and nuclear spins. The coupling of the electron spin subsystem to the longitudinal sound allows making the velocity of the latter close to that of the transverse acoustic waves, which provides an effective interaction between all components of the elastic field by means of the nuclear spin subsystem. We derive a three-component system of material and reduced wave equations describing this process and construct its soliton solutions in the form of stationary and breather pulses. Based on them, we study the peculiarities of the dynamics of the elastic field components and reveal the differences from the two-component model. The existence of two families of breathers is an important distinctive feature of the considered case.     

Resonance effects in the field of a stationary electromagnetic wave and a constant magnetic field

We analyze the interaction of a charged relativistic particle with the electromagnetic field given by a superposition of a stationary wave and a constant magnetic field. The refraction coefficient of the medium is taken different from one. We investigate the problem in the low-signal approximation. To test the results, we used computer simulation of the original system with two scales of the variation rate of the variables taken into account.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 64–71, January, 2005.     

Long Time Motion of NLS Solitary Waves in a Confining Potential

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schr?dinger equations with a confining, slowly varying external potential, V(x). A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential V(x) over a long time interval. Communicated by Rafael D. Benguria Submitted: March 7, 2005 Accepted: January 9, 2006     

Kink solutions of the classical anisotropic XY chain

We investigate the classical XY model, which is a lattice differential equation on the one-dimensional integer lattice. Depending on an anisotropy parameter γ and an external transverse field of strength λ, we prove the existence of stationary and travelling wave kink solutions.     

Cortical waves and post‐stroke brain stimulation

A neural field model with different activation and inhibition connectivity and response functions is considered. Stability analysis of a homogeneous in space solution determines the conditions of the emergence of stationary periodic solutions and of periodic travelling waves. Various regimes of wave propagation are illustrated in numerical simulations. The influence of external stimulation on the wave properties is investigated.     

Exact solutions of a two-fluid model of two-phase compressible flows with gravity

We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave.     

Waves in a Perfectly Conducting Fluid Filling a Half-Space

The constant, maximal, energy preserving boundary conditionsfor the equations of magnetohydrodynamics in a perfectly conductinghalf-space give rise to two essentially different selfadjointoperators in the case when the external magnetic field is orthogonalto the boundary and exactly one such operator when the externalfield is parallel to the boundary. Neither of these problemsadmits surface waves. For a normalized external field, the generalizedeigenfunction expansion is given below. It is shown that, inthe second case, the modes are not coupled by the boundary,while for only one boundary condition for the orthogonal fieldis the wave motion essentially that of free space (in the sensethat solutions are delivered by the group which determines solutionsfor the free space problem for special initial data). The Alvnwave in the parallel field case acts as a grazing wave. Asymptoticwave motion for perturbed problems (inhomogeneous media) isinvestigated as well as local decay of energy (this is not altogethertrivial, since the operators involved are never coercive evenoff their null spaces).     

Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings

We consider interaction of charged particles with an electromagnetic (electrostatic) low frequency wave propagating perpendicular to a uniform background magnetic field. The effects of particle trapping by the wave and further acceleration of a surfatron type are discussed in details. Method for this analysis based on the adiabatic theory of separatrix crossing is used. It is shown that particle can unlimitedly accelerate in the trapping in electromagnetic waves and energy of particle does not increase for the system with electrostatic wave.     

Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors

In this paper, we discuss a multi-dimensional stationary flow model for semiconductor devices which is based on the nonisentropic hydrodynamic equations with non-constant lattice temperature. We establish the existence of the irrotational non-thermal equilibrium subsonic steady state solutions with positive particle density and positive carrier temperature. The proofs are based on the Schauder fixed-point principle, Stampacchia’s truncation methods and the careful energy estimates. Meanwhile, we also show that these strong stationary solutions are unique.     

Localized minimizers of flat rotating gravitational systems

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.     

Travelling Waves with a Singularity in Magnetic Nanowires

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.     

Traveling wave solutions of a model of skin pattern formation in a singular case

We study traveling wave solutions to a system of four non‐linear partial differential equations, which arise in a tissue interaction model for skin morphogenesis. Under the assumption that the strength of attachment of the epidermis to the basal lamina is sufficiently large, we prove the existence and uniqueness (up to a translation) of traveling wave solutions connecting two stationary states of the system with the dermis and epidermis cell densities being positive. We discuss the problem of the minimal wave speed. Copyright © 2010 John Wiley & Sons, Ltd.     

Scattering of solitons for Dirac equation coupled to a particle

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and symplectic projection onto solitary manifold in the Hilbert phase space.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation

The asymptotic behaviors of solutions of an initial-boundary value problem for the generalized BBM equation with non-convex flux are discussed in this paper. It is proved that under the conditions of constant boundary data and small perturbation for the initial data, the global solutions exist and converge time-asymptotically to a stationary wave or the superposition of a stationary wave and a rarefaction wave. The proof is given by a technical L ²-weighted energy method.     

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with n_r 1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 83–111, April, 2005.     

On in‐out splitting of incident fields and the far‐field behaviour of Herglotz wavefunctions

It is easy to write down entire solutions of the Helmholtz equation: Examples are plane waves and Herglotz wavefunctions. We are interested in the far‐field behaviour of these solutions motivated by the following question: When is it legitimate to split the far field of such an entire solution into the sum of an incoming spherical wave and an outgoing spherical wave? We review the relevant literature (there are disjoint physical and mathematical threads), and then we answer the question for Herglotz wavefunctions, using a combination of the 2‐dimensional method of stationary phase and some explicit examples.