The whispering surface effect

The representation theory of symmetry groups, together with variational and functional-topological methods, are used in a two-dimensional formulation to investigate the waveguide properties of one-dimensionally periodic surfaces (OPS) and interfaces. It is established that all surfaces on which the Neumann condition is satisfied possess the waveguide property—they are open waveguides. This means that there are waves localized in the neighbourhood of the surface which propagate along it without attenuation—waveguide modes. It is shown that for any hard OPS there is always a transmission band of waveguide frequencies, localized in the neighbourhood of zero—the whispering surface effect. Anomalous oscillations localized around OPSs on which the Neumann condition is satisfied are observed and investigated. Examples of surfaces for which anomalous oscillations exist and others for which none exist are presented. It is proved that OPSs on which the Dirichlet condition holds do not have a transmission band for waveguide frequencies in the neighbourhood of zero, and for some frequency bands they do not have waveguide and anomalous properties. It is shown that one-dimensionally periodic interfaces of two media possess waveguide and anomalous properties, provided that the parameters satisfy certain relationships. It is established that if the interface has the waveguide property, then transmission band of frequencies will always exist localized in the neighbourhood of zero—the whispering interface effect. An example is presented in which anomalous oscillations are investigated, dispersion relations are derived and pass and stop bands for waveguide modes are determined.     

Geometric Aspects of Spatially Periodic Interfacial Waves

Periodic waves at the interface between two inviscid fluids of differing densities are considered from a geometric point of view. A new Hamiltonian formulation is used in the analysis and restriction of the Hamiltonian structure to space-periodic functions leads to an O -invariant Hamiltonian system. Motivated by the simplest O -invariant Hamiltonian system, the spherical pendulum, we analyze the properties of traveling waves, standing waves, interactions between standing and traveling waves (mixed waves) and time-modulated spatially periodic waves. A singularity in the bifurcation of traveling waves leads to a nonlinear resonance and this is investigated numerically.     

Nonstationary normal waves in a thin,curved waveguide in a multiray zone (uniform asymptotics)

In a thin waveguide with properties that vary along its course, the propagation of nonstationary normal waves in the presence of caustics for space-time rays is considered. The connection of the critical section of the waveguide with such caustics is determined. Uniform asymptotic formulas are obtained for the wave field in a multiray zone, and the passage into geometric rays outside a neighborhood of the caustics is traced.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 127–137, 1980.In conclusion, we note that the matrix machinery used in the present work is not connected with the special nature of the present problem and can always be applied when the initial Ansatz contains a collection of special functions which go over into one another under differentiation.     

Acoustic eigenoscillations near thin-walled obstacles in an annular cylindrical channel

Under analytical and numerical studies are the acoustic eigenoscillations near thinwalled obstacles in homogeneous circular cylindrical channels. The acoustic eigenoscillations are described by the Neumann problem for the Laplace operator. Using the representation theory of the symmetry groups in the solution space, it is shown that, for a large class of thin-walled obstacles in the circular channels, there always exists a pure point spectrum immersed into the continuous spectrum of the self-adjoint extension of the Laplace operator corresponding to the homogeneous Neumann problem. The dependencies are obtained of the eigenfrequencies on the geometric parameters of the thin-walled obstacles in a homogeneous cylindrical circular channel. The form of the eigenfunctions is investigated. The influence is discussed of the geometric characteristic of the domain on the frequencies, number, and form of eigenoscillations.     

A Geometric Construction of Traveling Waves in a Bioremediation Model

Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding to the motion of a biologically active zone, in which the microorganisms consume both substrate and acceptor. For certain values of the parameters, the traveling waves exist on a three-dimensional slow manifold within the five-dimensional phase space. We prove persistence of the slow manifold under perturbation by controlling the nonlinearity via a change of coordinates, and we construct the wave in the transverse intersection of appropriate stable and unstable manifolds in this slow manifold. We study how the TW depends on the half-saturation constants and other parameters and investigate numerically a bifurcation in which the TW loses stability to a periodic wave.     

Existence and uniqueness of generalized stationary waves for viscous gas flow through a nozzle with discontinuous cross section

In this paper we study the existence and uniqueness of the generalized stationary waves for one-dimensional viscous isentropic compressible flows through a nozzle with discontinuous cross section. Following the geometric singular perturbation technique, we establish the existence and uniqueness of inviscid and viscous stationary waves for the regularized systems with mollified cross section. Then, the generalized inviscid stationary waves are classified for discontinuous and expanding or contracting nozzles by the limiting argument. Moreover, we obtain the generalized viscous stationary waves by using Helly?s selection principle. However, due to the choices of mollified cross section functions, there may exist multiple transonic standing shocks in the generalized stationary waves. A new entropy condition is imposed to select a unique admissible standing shock in generalized stationary wave. We show that, such admissible solution selected by the entropy condition, admits minimal total variation and has minimal enthalpy loss across the standing shock in the limiting process.     

TRAVELING WAVES IN A BIOLOGICAL REACTION-DIFFUSION MODEL WITH STRONG GENERIC DELAY KERNEL AND NON-LOCAL EFFECT

In this paper,we consider the reaction diffusion equations with strong generic delay kernel and non-local effect,which models the microbial growth in a flow reactor.The existence of traveling waves is established for this model.More precisely,using the geometric singular perturbation theory,we show that traveling wave solutions exist provided that the delay is sufficiently small with the strong generic delay kernel.     

Conical diffraction by periodic structures: Variation of interfaces and gradient formulas

This paper studies the dependence of solutions to conical diffraction problems upon geometric parameters of non–smooth profiles and interfaces between different materials of diffractive gratings. This problem arises in the design of those optical devices to diffract time–harmonic oblique incident plane waves to a specified far–field pattern. We prove the stability of solutions and give analytic formulas for the derivatives of reflection and transmission coefficients with respect to Lipschitz perturbations of interfaces. These derivatives are expressible as contour integrals involving the direct and adjoint solutions of conical diffraction problems.     

Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

The linear stability of the solitary waves for the one‐dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.     

Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum

The spectra of open angular waveguides obtained by thickening or thinning the links of a thin square lattice of quantum waveguides (the Dirichlet problem for the Helmholtz equation) are investigated. Asymptotics of spectral bands and spectral gaps (i.e., zones of wave transmission and wave stopping, respectively) for waveguides with variously shaped periodicity cells are found. It is shown that there exist eigenfunctions of two types: localized around nodes of a waveguide and on its links. Points of the discrete spectrum of a perturbed lattice with eigenfunctions concentrated about corners of the waveguide are found.     

Numerical Method for Homoclinic and Heteroclinic Orbits of Neuron Models

A twisted heteroclinic cycle was proved to exist more than twenty- five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their traveling wave moving frame. The result implies the existence of infinitely many traveling front waves and infinitely many traveling back waves for the system. However efforts to numerically render the twisted cycle were not fruit- ful for the main reason that such orbits are structurally unstable. Presented here is a bisectional search method for the primary types of traveling wave solu- tions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo equations represent. The algorithm converges at a geometric rate and the wave speed can be approximated to significant precision in principle. The method is then applied for a recently obtained axon model with the conclusion that twisted heteroclinic cycle maybe more of a theoretical artifact.     

Plane harmonic wave propagation in three-dimensional composite media

An important characteristic of waves propagating through periodic materials is the existence of stop bands. A stop band implies the range of frequencies over which a medium completely reflects all incident waves and there is no transmission. Predicting stop band phenomena in periodic materials is regarded as the first step toward designing composite microstructures capable of propagating energy in a predetermined manner. In this paper a global–local modeling methodology previously proposed by the authors is used to successively predict stop bands in three-dimensional composite media. Numerical results reveal that the first stop band of the considered microstructures occurs where an acoustic shear mode veers with the lowest optical branch of the same symmetry class.     

Reverse time migration for reconstructing extended obstacles in planar acoustic waveguides

We propose a new reverse time migration method for reconstructing extended obstacles in the planar waveguide using acoustic waves at a fixed frequency. We prove the resolution of the reconstruction method in terms of the aperture and the thickness of the waveguide. The resolution analysis implies that the imaginary part of the cross-correlation imaging function is always positive and thus may have better stability properties.Numerical experiments are included to illustrate the powerful imaging quality and to confirm our resolution results.     

Spectrally Stable Encapsulated Vortices for Nonlinear Schr?dinger Equations

Summary. A large class of multidimensional nonlinear Schrodinger equations admit localized nonradial standing-wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (nonlocalized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeroes of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.     

Collective vibrations in superconductors with low carrier density and overlapping energy bands on the fermi surface

Collective vibrations in two-band superconductors with low carrier densities are studied for the case described by the BCS superconductivity theory and for the case where localized pairs form a Bose condensate (the Schaffroth theory of superconductivity). It is shown that in both cases, there exists an acoustic mode of collective vibrations and an exciton-type collective mode caused by interband electron-electron interactions. For systems with low carrier densities, the frequencies of the collective vibrations are significantly influenced by the mixing of the phase and amplitude fluctuations of the order parameters for different bands. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 162–175, January, 1997.     

Soliton-like lamb waves

The velocity and polarization of acoustic Lamb waves, propagating in the directions of elastic symmetry of single-layer and double-layer anisotropic media at vanishingly low frequencies (soliton-like waves), are investigated. The method of fundamental matrices is used to construct solutions. The conditions for soliton-like Lamb waves to exist are analysed.     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves

The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by Phillips (1960) is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips’ four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple-variables are properly defined with clear physical meanings.     

On the piezoelectric shear effect in ultrasonic motors

A new actuator for piezoelectric ultrasonic motors (USM) using the d₁₅ effect was conceived. Whereas the piezoelectric d₃₃ and d₃₁ effects are normally used in commercial motors, there exist hardly any USM based on shear actuation. The actuator is a piezoelectric block polarized in axial direction and electroded circumferentially with four electrodes. The suitable superposition of two standing waves generates ultrasonic traveling waves in the actuator, which drives the rotor. The dimensions of the actuator are optimized with respect to the dynamic piezoelectric coupling factor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.