Thermoelasticity with second sound—exponential stability in linear and non‐linear 1‐d

We consider linear and non‐linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave‐like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic–parabolic thermoelasticity is given. For Dirichlet type boundary conditions—rigidly clamped, constant temperature—the global existence of small, smooth solutions and the exponential stability are proved for a non‐linear system. Copyright © 2002 John Wiley & Sons, Ltd.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

Integrable models of the dynamics of longitudinal-transverse acoustic pulses in a paramagnetic crystal

We study the interaction between longitudinal-transverse acoustic pulses and a system of paramagnetic impurities with the effective spin S = 1 in a statically deformed crystal. We show that the dynamics of a pulse propagating at an arbitrary angle to the static-deformation direction and of the effective spins satisfy the modified reduced Maxwell-Bloch equations and, if the spectrum of the acoustic pulse overlaps the quantum transitions between spin sublevels, the modified sine-Gordon equation. These equations generalize the well-known models in the theory of the inverse scattering method and in the theory of self-induced transparency and also belong to the class of integrable equations. Analyzing soliton solutions shows that the pulse-medium interaction reveals some qualitatively new features in these models compared with the cases of purely transverse or purely longitudinal acoustic fields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 228–247, May, 2007.     

Mathematical simulation of light pulse propagating within a microring resonator system and applications

This paper presents the very fascinating simulation results of light pulse traveling within a ring resonator system that have shown the unexpected results with various applications. The design system consists of a nonlinear microring/nanoring resonator system incorporating an add/drop filter. The proposed fabricated material used is InGaAsP/InP, which can provide the required output behaviors. Three different forms of input light pulses are Gaussian pulse, dark and bright soliton, whereas the suitable simulation parameters are input power, pulse width, ring radii and the material refractive indices. Three different forms of the results have been interpreted, whereas the dominants behaviors are such as Gaussian soliton, multisoliton and tunable dark soliton are described, and the potential applications for new laser sources, new communication bands and dynamic optical tweezers have been discussed.     

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a photonic crystal with combined nonlinearity

A conservative finite-difference scheme is constructed for the problem of propagation of a light pulse in a one-dimensional nonlinear photonic crystal with combined nonlinearity. The invariants of the corresponding differential problem and their difference analogues are given. The scheme is compared with those based on the widespread splitting method. For combined cubic and quadratic nonlinearity in photonic crystal layers, it is shown that the classical splitting method is ineffective, since it requires time steps that are smaller by one or more orders of magnitude. The finite-difference scheme proposed conserves the propagation invariants, which cannot be achieved for splitting schemes even on considerably finer grids. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes as applied to the simulation of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The simulation is based on the approach proposed by the authors for the given class of problems.     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Multimode systems of nonlinear equations: Derivation,integrability, and numerical solutions

We consider the propagation of electromagnetic pulses in isotropic media taking a third-order nonlinearity into account. We develop a method for transforming Maxwell’s equations based on a complete set of projection operators corresponding to wave-dispersion branches (in a waveguide or in matter) with the propagation direction taken into account. The most important result of applying the method is a system of equations describing the one-dimensional dynamics of pulses propagating in opposite directions without accounting for dispersion. We derive the corresponding self-action equations. We thus introduce dispersion in the media and show how the operators change. We obtain generalized Schäfer-Wayne short-pulse equations accounting for both propagation directions. In the three-dimensional problem, we focus on optic fibers with dispersive matter, deriving and numerically solving equations of the waveguide-mode interaction. We discuss the effects of the interaction of unidirectional pulses. For the coupled nonlinear Schrödinger equations, we discuss a concept of numerical integrability and apply the developed calculation schemes.     

Transient analysis of a potential pulse in a piezoelectric (622) crystal plate

Transient analysis of the wave propagation due to an electric potential pulse concentrated at a point on the boundary of an infinite piezoelectric plate of the crystal class (622) resting on an infinite elastic medium is considered. The electric potential is studied for an exponential pulse and step pulse. Numerical results are obtained for theβ-quartz plate that rests on aluminium half-space.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

Periodic Optical Pulses in Nonlinear Optical Fibers

We discuss the problem of transmitting polarized pulses along optical fibers with variable dispersion. The dissipation and mean dispersion are assumed to be zero, which allows using the model of the vector nonlinear Schrödinger equation. We consider an optical fiber consisting of arms of equal length, which is assumed to be large. We propose an asymptotic recursive procedure for calculating the amplitude and the phase of an optical pulse propagating along the optical cable with variable dispersion.     

A Numerical Study of the Light Bullets Interaction in the (2+1) Sine-Gordon Equation

The propagation and interaction in more than one space dimension of localized pulse solutions (so-called light bullets) to the sine-Gordon [SG] equation is studied both asymptotically and numerically. Similar solutions and their resemblance to solitons in integrable systems were observed numerically before in vector Maxwell systems. The simplicity of SG allows us to perform an asymptotic analysis of counterpropagating pulses, as well as a fully resolved computation over rectangular domains. Numerical experiments are carried out on single pulse propagation and on two pulse collision under different orientations. The particle nature, as known for solitons, persists in these two space dimensional solutions as long as the amplitudes of initial data range in a finite interval, similar to the conditions on the vector Maxwell systems.     

On Impulse Distortion in Dispersive Media

Pulse distortion is due to nonlinear wave number functions of the frequency. For the case of smooth nonlinear wave numbers, a sequence of approximations to the propagation operator which carries the incident pulse to the distorted output pulse is set forth. When applied to the actual incident pulse, they allow the study of the development of the distortion with (a) increasing travel length in the propagation medium and (b) also with respect to the properties of the propagation medium. Of special interest are Gaussian incident pulses for which the distorted pulse form is more concisely representable. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transition Between Linear and Exponential Propagation in Fisher-KPP Type Reaction-Diffusion Equations

We study the Fisher-KPP equation with a fractional Laplacian of order α ∈ (0, 1). We know that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α ∈ (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order -ln (1 ? α), the propagation is linear, and becomes exponential as soon as the time exceeds a large multiple of -ln (1 ? α).     

Nonlinear supratransmission in a discrete nonlinear electrical transmission line: Modulated gap peak solitons

We examine the propagation conditions of modulated gap signal voltage in a one-dimensional nonlinear discrete electrical transmission line where nonlinear capacitors are introduced both in their series and shunt branches. Especially, we consider that the propagating signal voltage frequency belong to the forbidden band zones. Using the exact discrete equations of the network and the extended nonlinear Schrödinger equation, the threshold value of the signal amplitude over which pulse soliton is emitted in the network is found. We show that when the signal amplitude exceeds a particular value, the critical value, peaked soliton can be emitted by the network due to the nonlinear dispersive elements of the series branches. Numerical simulations, confirming the exactness of the analytical analysis are performed on the exact equations of the network.     

Finite‐difference time‐domain simulation of acoustic propagation in heterogeneous dispersive medium

Accurate modeling of pulse propagation and scattering is a problem in many disciplines (i.e., electromagnetics and acoustics). It is even more tenuous when the medium is dispersive. Blackstock [D. T. Blackstock, J Acoust Soc Am 77 (1985) 2050] first proposed a theory that resulted in adding an additional term (the derivative of the convolution between the causal time‐domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. Thus deriving a modified wave equation applicable to either linear or nonlinear propagation. For the case of an acoustic wave propagating in a two‐dimensional heterogeneous dispersive medium, a finite‐difference time‐domain representation of the modified linear wave equation can been used to solve for the acoustic pressure. The method is applied to the case of scattering from and propagating through a 2‐D infinitely long cylinder with the properties of fat tissue encapsulating a cyst. It is found that ignoring the heterogeneity in the medium can lead to significant error in the propagated/scattered field. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Nonlinear Resonances Leading to Strong Pulse Attenuation in Granular Dimer Chains

We study nonlinear resonances in granular periodic one-dimensional chains. Specifically, we consider a diatomic (“dimer”) chain composed of alternating “heavy” and “light” spherical beads with no precompression. In a previous work (Jayaprakash et al. in Phys. Rev. E 83(3):036606, 2011) we discussed the existence of families of solitary waves in these systems that propagate without distortion of their waveforms. We attributed this dynamical feature to “antiresonance” in the dimer that led to the complete elimination of radiating waves in the trail of the propagating solitary wave. Antiresonances were associated with certain symmetries of the velocity waveforms of the dimer beads. In this work we report on the opposite phenomenon: the break of waveform symmetries, leading to drastic attenuation of traveling pulses due to radiation of traveling waves to the far field. We use the connotation of “resonance” to describe this dynamical phenomenon resulting in maximum amplification of the amplitudes of radiated waves that emanate from the propagating pulse. Each antiresonance can be related to a corresponding resonance in the appropriate parameter plane. We study the nonlinear resonance mechanism numerically and analytically and show that it can lead to drastic attenuation of pulses propagating in the dimer. Furthermore, we estimate the discrete values of the normalized mass ratio between the light and heavy beads of the dimer for which resonances are realized. Finally, we show that by adding precompression the resonance mechanism gradually degrades, as does the capacity of the dimer to passively attenuate propagating pulses.     

Amplification and compression of Ti:sapphire femtosecond pulses at high-repetition-rate

The experiment of the generation and amplification of femetosecond Ti:sapphire laser pulse at high repetition rate is reported. The laser pulses with minimum pulsewidth 15 fs, maximum spectrum width of 80 nm, average power of 200 mW are generated from a home-built self-mode-locked Ti:sapphire laser. As a seed pulse which is selected from the oscillator, the laser pulse is further amplified by using chirped-pulse-amplification technology in a Ti:sapphire amplifier from which a kind of pulses with single-pulse-energy of 100 uj, pulsewidth after compressing of 50 fs at 5 kHz repetition rate are produced. The system design and experimental results are discussed. Project supported by the National “Climbing Project” of China.     

Split fields and direction of propagation for the solution to first-order systems of equations

The standard wave-splitting approach for the wave equation in inhomogeneous media is first reexamined. Next, by analogy with the theory of wave propagation through singular surfaces, a characterization is given for a function in space-time to represent a wave propagating in a direction. The condition is applied in connection with a simple example and found to be quite restrictive. The same problem is then considered in the Fourier-transform domain where the unknown function is an n-tuple satisfying a system of ordinary differential equations. The condition for propagation in a direction is established for the Fourier components. Next, some physical problems are considered which are expressed by partial differential equations or by integro-differential equations. The associated first-order system of equations is examined in terms of the eigenvalues of a matrix. This shows that, for any eigenvalue, the direction of propagation may change with the frequency and that arguments about the dominance of the principal part of the operator may cease to hold.     

Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term

In this work, we consider a linear thermoelastic laminated timoshenko beam with distributed delay, where the heat conduction is given by cattaneoâs law. we establish the well posedness of the system. For stability results, we prove exponential and polynomial stabilities of the system for the cases of equal and nonequal speeds of wave propagation.     

On the stability of Timoshenko-type systems with internal frictional dampings and discrete time delays

In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain under Dirichlet–Dirichlet or Dirichlet–Neumann boundary conditions with one or two discrete time delays and one or two internal frictional dampings. First, we show that the system is well posed in the sens of semigroup theory. Second, we prove the exponential stability regardless to the speeds of wave propagation of the system if the weights of the time delays are smaller than the ones of the corresponding dampings, respectively. However, when the weight of one time delay is not smaller than the one of the corresponding damping, we prove the exponential stability in case of equal-speed wave propagation, and the polynomial stability in the opposite case.