Nonlinear Propagation Problems Research Articles

Burgers equation and convexification method: Propagation of wave packets and narrowband noise.

The Burgers equation is a standard model for the propagation of progressive finite amplitude waves in a lossy medium. This paper is mainly devoted to the one-dimensional Burgers equation and its solution deduced from a convexification method. Its starting point is the Hopf-Cole solution in the inviscid limit, then the Legendre-Fenchel transform and the convex envelope construction are employed to obtain the single-value waveform. A fractional step method is used to numerically solve the Burgers equation in a segregated manner (nonlinearity and dissipation). Numerical results are obtained for nonlinear propagation of a wave packet with a Gaussian envelope and a narrowband noise (deterministic and non-deterministic signals). The convexification method provides a tool to solve the problems of nonlinear propagation, especially in the case of noise propagation with the formation and interaction (coalescence) of a large number of discontinuities (shocks).

A new class of diagonally implicit Runge–Kutta methods with zero dissipation and minimized dispersion error

In this work, we propose a new class of A-stable diagonally implicit four stage Runge–Kutta (R–K) methods with minimal dissipation and optimally low dispersion error. These schemes obtained by minimizing both amplification and phase error enjoy fourth order of accuracy and are suitable for stiff systems. Emphasis here is to outline an algorithm that can be used to develop diagonally implicit R–K methods of diverse stages having low-dissipation low-dispersion virtues while retaining, to a large extent, inherent stability and high accuracy. This algorithm is subsequently applied to propose two, three and four stage diagonally implicit R–K schemes. One and two dimensional, linear and non-linear propagation problems are numerically tackled at a relatively higher CFL number and a comprehensive comparison is carried out with other stable diagonally implicit schemes available in the literature to exhibit benefits of optimization.

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as a truly meshfree method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation was deployed which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional defined as the sum of squared residuals of the differential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-definite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and verified by a set of benchmark problems. The results indicate the ability of proposed method to efficiently and effectively solve the linear and non-linear propagation problems.

New ducting model for analyzing the Gaussian beam propagation in nonlinear Kerr media and its application to spatial self-phase modulations

Knowing the Gaussian beam parameters, such as its radius of curvature and spot size during propagation in nonlinear Kerr media, is of paramount importance in describing the observable far-field diffraction ring patterns as well as in design and stability analysis of Kerr-lens mode-locked resonators. Specifically, the sign of the beam radius of curvature after exiting these media has been proposed to be of assistance in recognizing their optical nonlinearity sign through determining the type of diffraction ring pattern in the far field. In order to be able to trace the evolution of the beam parameters in the Gaussian beam formalism, we have used the common aberration-free theory. We have shown that the nonlinear propagation problem of a fundamental Gaussian beam in a Kerr medium with an intensity-dependent index of refraction can be handled by assuming a ducting index profile along the propagation direction. Knowing the familiar ABCD matrix of a duct, the evolution of the mentioned beam parameters can be traced during propagation using the ABCD law in Gaussian beam theory. We have validated our ducting model by comparing its results with the outcomes of one widely used and accepted model which has been known to yield consistent results when electronic optical nonlinearity prevails. We have shown that when thermal optical nonlinearity is dominant, as in diffraction ring observation experiments, our ducting model yields sensible results and should be used. Our model predicts that when the sign of the thermal nonlinearity and the beam radius of curvature on the entrance plane of the medium are positive, the sign of the beam radius of curvature on the exit plane may have either sign, depending on the medium thickness used in the experiment. Hence, two types of diffraction ring pattern may be obtained using the same medium with two different thicknesses and this may cast doubt on the validity of the methods proposing the detection of the optical nonlinearity signs by observing these patterns. We have proposed a simple procedure for experimentally obtaining the two different types of diffraction pattern from the same medium.

General approach to spatiotemporal modulational instability processes

In this article, we derive the general exact solution of the modulation instability gain. The solution described here is valid for 1-D, 2-D, and 3-D cases considering any temporal response function of the medium and with possible higher order Kerr nonlinearities. In particular, we show that the gain induced by modulation instability is initial condition dependent, while the usual calculations do not lead to such a dependence. Applications for current and high-interest nonlinear propagation problems, such as 1-D optical fiber propagation with delayed Raman response and 2-D filamentation in gases, are investigated in detail. More specifically, we demonstrate that the 2-D model of filamentation based on the balance between higher order Kerr terms leads to a modulation instability window. The impact of both self-steepening and space-time defocusing effects is also highlighted. Finally, we discuss the influence of the finite-time response of the different order electronic Kerr effects on the growth of the expected modulation instability bands.

On perturbative solutions for nonlinear waves in inhomogeneous materials

Complex phenomena of wave-prestress and wave-material interactions are studied. Model problems of nonlinear propagation of longitudinal waves in materials (i) with inhomogeneous prestress and (ii) with weakly variable physical properties are posed. In both problems two small parameters of different physical nature appear. The problems are solved using the perturbation technique. The influence of the values of two small parameters on the content of terms of perturbative solutions is analysed. The outcome facilitates utilization of perturbative solutions in composing algorithms for ultrasonic nondestructive testing of materials with variable properties.

Non-linear Young's double-slit experiment

The Young's double slit experiment is recreated using intense and short laser pulses. Our experiment evidences the role of the non-linear Kerr effect in the formation of interference patterns. In particular, our results evidence a mixed mechanism in which the zeroth diffraction order of each slit are mainly affected by self-focusing and self-phase modulation, while the higher orders propagate linearly. Despite of the complexity of the general problem of non-linear propagation, we demonstrate that this experiment retains its simplicity and allows for a geometrical interpretation in terms of simple optical paths. In consequence, our results may provide key ideas on experiments on the formation of interference patterns with intense laser fields in Kerr media.

Low-dissipation and low-dispersion fourth-order Runge–Kutta algorithm

An optimized explicit low-storage fourth-order Runge–Kutta algorithm is proposed in the present work for time integration. Dispersion and dissipation of the scheme are minimized in the Fourier space over a large range of frequencies for linear operators while enforcing a wide stability range. The scheme remains of order four with nonlinear operators thanks to the low-storage algorithm. Linear and nonlinear propagation problems are finally solved to illustrate the accuracy of the present Runge–Kutta scheme.

Propagation of Ultrasonic Waves in Nonlinear Multilayered Media

In the framework of the Local Interaction Simulation Approach (LISA), iteration equations are derived for the simulation of the propagation of ultrasonic waves in nonlinear multilayered 1-D media. The numerical results are compared with analytical solutions obtained in a perturbation treatment. The method is then applied to study several problems of nonlinear propagation, including the distortion of pulses, the formation of shock waves and, in the case of cw's, anharmonic effects and coupling between extended and localized vibration modes.

PROPAGATION OF ULTRASONIC WAVES IN NONLINEAR MULTILAYERED MEDIA

In the framework of the Local Interaction Simulation Approach (LISA), iteration equations are derived for the simulation of the propagation of ultrasonic waves in nonlinear multilayered 1-D media. The numerical results are compared with analytical solutions obtained in a perturbation treatment. The method is then applied to study several problems of nonlinear propagation, including the distortion of pulses, the formation of shock waves and, in the case of cw's, anharmonic effects and coupling between extended and localized vibration modes.

Nonlinear interaction of longitudinal waves in an inhomogeneously predeformed elastic medium

Interaction of longitudinal waves in an elastic medium with inhomogeneous plane prestrain is investigated theoretically. Simultaneous propagation of two waves excited by tractions on opposite boundaries of the medium (structural element) is considered. An analytical solution to the one-dimensional problem of nonlinear propagation and interaction of waves is derived. Harmonic wave propagation in the prestrained medium is studied in detail. Influence of the prestrain on the sine-wave induced stress is cleared up on the boundaries of the medium. A method to estimate the parameters of the predeformed state is proposed.

Nonlinear waves in characterization of inhomogeneous elastic material

The nonlinear longitudinal wave propagation in weakly inhomogeneous elastic material with nonlinear physical properties is investigated theoretically. The analytical solution to the one-dimensional problem of nonlinear propagation of the wave with an arbitrary smooth initial profile is derived. The solution establishes the relations between the variable physical properties of the material and the wave characteristics. The possibility to utilize this solution in inhomogeneous elastic material characterization is considered on the basis of the model problem of sine-wave propagation. The material properties are described by polynomials with respect to the spatial coordinate. The dependencies of the sine-wave characteristics on the variable physical properties of the material are illustrated in numerous plots. The algorithm to nondestructive evaluation of the linearly variable properties of the elastic material is proposed.

Modelling nonlinear and dispersive propagation problems by using the TLM method

The transmission-line modelling (TLM) method is extended to the analysis of dispersive and nonlinear propagation problems, significantly increasing the range of application of the method. Dispersion and nonlinearities are incorporated into the TLM scheme by loading with appropriate stubs. The modelling of nonlinear effects is illustrated by the incorporation of a nonlinear capacitor model, using the nonlinear discrete transform method, into the TLM method. Correct operation and energy conservation of the nonlinear discrete-transform method is discussed. The method is applied to the analysis of soliton propagation on a lumped component transmission line, and also to the self-focusing of femtosecond pulses in a Kerr medium. A dispersive stub, designed to model electromagnetic waves propagating in a plasma, is introduced which may be used to load one, two and three-dimensional TLM field nodes.

Influence of the maximum space-charge field on the rate of self-bending of the path of a speckle beam in photorefractive crystals

A self-similar solution of the problem of nonlinear propagation of a speckle beam was obtained for photorefractive crystals with a diffusion-type response. The solution allows for the finite value of the maximum space-charge field. It is shown that this factor modifies the profile of the angular spectrum of the self-similar solution and increases the radius of curvature of the path compared with the case of an unrestricted field of a diffusion grating.

Coupled nonlinear stationary waves in a plasma

Coupled nonlinear waves in a plasma can often be expressed in terms of two nonlinear coupled ordinary second order differential equations by making the plane wave ansatz. In general these are quite difficult to solve unless some further drastic assumptions are made, e.g. one of the waves is treated as a driven wave. However, considerable progress can be made if one constant of motion is found. In this case the coupled stationary wave problem can be reduced to the standard classical mechanics problem of a nonlinear Hamiltonian with two degrees of freedom. Such Hamiltonians are known to generate a rich variety of nonlinear solutions which may be periodic, quasiperiodic or sometimes even ‘stochastic’. We obtain this broad range of solutions for the specific problem of nonlinear propagation of relativistically coupled electromagnetic and Langmuir waves in a cold plasma. The solutions (or ‘particle orbits’) are displayed in a Poincaré surface of section plots where they are classified and appropriately related to various approximate solutions discussed before.

Surface acoustic waves of finite amplitude excited by a monochromatic line source

A technique for obtaining a uniformly valid solution to the problem of nonlinear propagation of surface acoustic waves excited by a monochromatic line source is presented. The method of solution is an extension of the method of strained coordinates wherein both the dependent and independent variables are expanded in perturbation series. A special transformation is proposed for the independent variables so as to make the expansions uniformly valid and also to satisfy all the boundary conditions. This perturbation procedure, carried out to the second order, yields a solution containing a second harmonic surface wave whose amplitude and phase exhibit an oscillatory variation along the direction of propagation. In addition, the solution also contains a second harmonic bulk wave of constant amplitude but varying phase propagating into the medium.

EXPERIMENTAL EVIDENCE OF DISTORTED RIPPLING PROFILES OF FINITE AMPLITUDE ACOUSTIC SUFACE-WAVES

Waveform distortion of acoustic surface waves is experimentally analyzed through testing the modification of rippling profiles. An optical probing method has been used, which allows one to measure the phase shift between fundamental and second harmonic through the asymmetries of the light diffracted in second orders by the distorted waves. Several configurations of the rippling profiles have been investigated of surface waves propagating on LiNbOo, LiTa03, Bii2Ge020 a SiOg. Cases have been found where phase relation between fundamental and second harmonic waves is such as to produce straight forward or backward sloping profiles. 1.INTRODUCTION. It is well known that the growing of harmonic along the propagation path of finite amplitude acoustic bulk waves in condensed media produces characteristic sawtooth distortions of the velocity profiles. The waveforms develop along the propagation direction into forward or backward sloping waves. These most stable configurations correspond to the two allowed phase relations between fundamental and second harmonic waves, which differ by a IT angle one with respect to the other. In case of surface waves, the inhomogeneous nature causes various waveform distortion to be expected. The final configuration of the wave, indeed, depends upon the relative amplitudes and phases with which nonlinear partial waves grow along the propagation direction. In the present work, waveform distortion of acoustic surface waves is experimentally analyzed through testing the modification of the rippling profiles along the propagation path. An optical method has been used, which allows one to measure the phase shift between fundamental and second harmonic through the asymmetries of the light diffracted into second orders by the distorted waves. 2. THE NON-LINEAR PROBLEM. The analytical approacn to tne problem of nonlinear propagation of acoustic surface waves has been variously attempted in recent times, through either some kind of phenomenological approach /1,2/ or simplifying assumptions /3,4/. A number of papers by Stegeman et al. /5/ recently obtained valuable theoretical results which match very closely the experimental date available. The analytical problem is to solve the nonlinear equation of motion : subjected to the stress-free boundary condition on the propagation surface. Here, p is the density of the medium, u the 'Hsplacer'ent vector and a., = ' ' Tk = 3U/3(3ui/axk), with U density of the internal energy. To our purpose, we presently discuss the problem in a very simplified qualitative way, by referring to the characteristics approach proper to a wide range of problems dealing with wave propagation. According to it, and limiting our attention first to the case of bulk waves in isotropic media, a family of curves can be traced on the plane (x,t) which are called characteristic curves and have the property that the acoustic perturbation is constant along anyone of them /6/. The slope of the curves gives the velocity with which the perturbation propagates at any point in space and instant of time. The curves run as parallel straight lines in case of linear propagation and may have different inclinations otherwise. In the former case, one obtains the well known result JOURNAL DE PHYSIQUE Colloque CQ, supplement au n° 11, tome 40, novembre 1979, page C8-249 Resume. La distorsion des ondes acoustiques de Rayleigh a ete experimentalement etudiee en analysant l'asymetrie du spectre de la lumiere diffractee produit par les changements du profil de deformation de la surface sur laquelle se propagent les ondes. On presente ici une methode optique qui a permis la determination experiementale de la relation de phase existant entre l'onde a presuence fondamentale et l'onde a la frequence de l'harmonique 2 pour plusieurs echantillons de cristaux piezo-electriques (LiNb03, LiTa03, Bi'i2Ge020 et Si02)• Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979843

The application of Gaussian beam. Formalism to optical propagation in nonlinear media

The nonlinear propagation problem of optical beams in media with an intensity dependent indices of refraction is solved in the case of a fundamental gaussian beam. This solution is made possible by assuming everywhere along the propagation path a quadratic index profile. The quadratic constant is obtained self-consistently from the second term in a Taylor expansion of the local gaussian intensity. The solution constitutes an extension of the well known ABCD formalism to nonlinear propagation.

Second harmonic generation in elastic surface waves on an isotropic solid

A procedure for solving the problem of non-linear propagation of elastic surface waves is given. An expression for the particle displacement of the second harmonic is obtained. It is shown that the amplitude of the harmonic increases linearly with distance and time and is proportional to the square of the amplitude of the fundamental frequency wave.

Use of impulsive sound in acoustical modeling

Impulsive sound has several advantages when used for acoustical scale modeling. These include elimination of the need for anechoic treatment to the laboratory walls, the identification of major paths of propagation, and the ability to correct for excess attenuation due to nonideal properties of the medium. When the impulsive sound is generated by a spark, the generation of the sound wave is inherently nonlinear, tending to follow the rules for blast wave production rather than the generation of sound by heat addition. Over certain distances and range of frequencies, the propagation is nonlinear, and a simple modification to geometric spreading appears to suffice. In the complementary linear regime, ordinary absorption rules appear to be appropriate. The conclusion from our studies is that the spark source is a useful and valuable sound source for modeling work in spite of problems of nonlinear propagation, electromagnetic interference, and high-voltage control. [Work supported by RANN program, NSF, and Department of Airports, City of Los Angeles.]