Solutions For Quasilinear Research Articles

Maximum modulus principles for radial solutions of quasilinear and fully nonlinear singular P.D.E's

We obtain maximum modulus principles for solutions to some quasilinear and fully nonlinear ODEs and discuss their applications to quasilinear PDEs involving $p$-Laplacian. Our approach is convenient to deal with singular PDEs. Its idea can be tracked back to the old theory by Szego on orthogonal polynomials.

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.

Modeling of finite-amplitude sound beams: second order fields generated by a parametric loudspeaker

The nonlinear interaction of sound waves in air has been applied to sound reproduction for audio applications. A directional audible sound can be generated by amplitude-modulating the ultrasound carrier with an audio signal, then transmitting it from a parametric loudspeaker. This brings the need of a computationally efficient model to describe the propagation of finite-amplitude sound beams for the system design and optimization. A quasilinear analytical solution capable of fast numerical evaluation is presented for the second-order fields of the sum-, difference-frequency and second harmonic components. It is based on a virtual-complex-source approach, wherein the source field is treated as an aggregation of a set of complex virtual sources located in complex distance, then the corresponding fundamental sound field is reduced to the computation of sums of simple functions by exploiting the integrability of Gaussian functions. By this result, the five-dimensional integral expressions for the second-order sound fields are simplified to one-dimensional integrals. Furthermore, a substantial analytical reduction to sums of single integrals also is derived for an arbitrary source distribution when the basis functions are expressible as a sum of products of trigonometric functions. The validity of the proposed method is confirmed by a comparison of numerical results with experimental data previously published for the rectangular ultrasonic transducer.

Form Drag Caused by Topographically Forced Waves in a Barotropic β Channel: Effect of Higher Mode Resonance

The relationship between form drag and the zonal mean velocity of steady states is investigated in a very simple system; a barotropic quasi-geostrophic β channel with sinusoidal topography. When a steady solution is calculated by the modified Marquardt method, keeping the zonal mean velocity constant as a parameter, the characteristic of the solution changes at a phase speed of a wave with a wavenumber higher than that of the bottom topography. For velocities smaller than this critical value, there exists a stable quasi-linear solution similar to the linear solution. For larger velocities, there exist three solutions whose form drag is very large. In addition, the resonant velocity of the mode, whose wavenumber is the same as the bottom topography, has no effect on these solutions. When the quiescent fluid is accelerated by a constant wind stress, acceleration stops around the critical velocity for a wide range of the wind stress. If the wind stress is too large for the acceleration to stop, the zonal current speed continues to increase infinitely. It is implied that the zonal velocity of equilibrium is mainly determined, not by the wind stress, but by the amplitude of the bottom topography and the viscosity coefficient.

Solutions for Quasilinear Schrödinger Equations via the Nehari Method

For a class of quasilinear Schrödinger equations we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method.

Quasilinear Approximation and WKB

Quasilinear solutions of the radial Schrodinger equation for different potentials are compared with corresponding WKB solutions. For this study, the Schrodinger equation is first cast into a nonlinear Riccati form. While the WKB method generates an expansion in powers of $\hbar$ , the quasi-linearization method (QLM) approaches the solution of the Riccati equation by approximating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameters. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning-point singularities. The first QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of different parameters, and the influence of their variation on the boundedness or unboundedness of a critically stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.

Moderately nonlinear ultrasound propagation in blood-mimicking fluid

In medical diagnostic ultrasound (US), higher than-in-water nonlinearity of body fluids and tissue usually does not produce strong nonlinearly distorted waves because of the high absorption. The relative influence of absorption and nonlinearity can be characterized by the Gol'dberg number Γ. There are two limiting cases in nonlinear acoustics: weak waves (Γ < 1) or strong waves (Γ >> 1). However, at diagnostic frequencies in tissue and body fluids, the nonlinear effects and effects of absorption more likely are comparable (Gol'dberg number Γ ≈ 1). The aim of this work was to study the nonlinear propagation of a moderately nonlinear US second harmonic signal in a blood-mimicking fluid. Quasilinear solutions to the KZK equation are presented, assuming radiation from a flat and geometrically focused circular Gaussian source. The solutions are expressed in a new simplified closed form and are in very good agreement with those of previous studies measuring and modeling Gaussian beams. The solutions also show good agreement with the measurements of the beams produced by commercially available transducers, even without special Gaussian shading. (E-mail: vince@bme.ri.ccf.org)

Quasilinear theory for the nonlinear Schrödinger equation with periodic coefficients

The nonlinear Schrodinger equation with periodic coefficients is analyzed under the condition of large variation in the local dispersion. The solution after n periods is represented as the sum of the solution to the linear part of the nonlinear Schrodinger equation and the nonlinear first-period correction multiplied by the number of periods n. An algorithm for calculating the quasilinear solution with arbitrary initial conditions is proposed. The nonlinear correction to the solution for a sequence of Gaussian pulses is obtained in the explicit form.

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Directional dependence of nonlinear surface acoustic waves in the (001) plane of cubic crystals.

Spectral evolution equations are used to perform analytical and numerical studies of nonlinear surface acoustic waves in the (001) plane of a variety of nonpiezoelectric cubic crystals. The basic theory underlying the model equations is outlined, and quasilinear solutions of the equations are presented. Expressions are also developed for a characteristic length scale for nonlinear distortion and a nonlinearity coefficient. A time-domain equation corresponding to the spectral equations is derived. Numerical calculations based on measured second- and third-order elastic constants taken from the literature are performed to predict the evolution of initially monofrequency surface waves. Nonlinearity matrix elements that indicate the coupling strength of harmonic interactions are shown to provide a useful tool for characterizing waveform distortion. The formation of compression or rarefaction shocks can be strongly dependent on the direction of propagation, and harmonic generation is suppressed or increased in certain directions.

Solution of the unsaturated soil moisture equation using repeated transforms

AbstractAn alternative method of solution for the linearized ‘theta‐based’ form of the Richards equation of unsaturated flow is developed in two spatial dimensions. The Laplace and Fourier transformations are employed to reduce the Richards equation to an ordinary differential equation in terms of a transformed moisture content and the transform variables, s and ξ. Separate analytic solutions to the transformed equation are developed for initial states which are either in equilibrium or dis‐equilibrium. The solutions are assembled into a finite layer formulation satisfying continuity of soil suction, thereby facilitating the analysis of horizontally stratified soil profiles. Solution techniques are outlined for various boundary conditions including prescribed constant moisture content, prescribed constant flux and flux as a function of moisture change. Example solutions are compared with linearized finite element solutions. The agreement is found to be good. An adaptation of the method for treating the quasilinearized Richards equation with variable diffusivity is also described. Comparisons of quasilinear solutions with some earlier semi‐analytical, finite element and finite difference results are also favourable. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Nonlinear Saturation of Topographically Forced Rossby Waves in a Barotropic Model

A quasigeostrophic barotropic model is used to examine the nonlinear saturation of forced Rossby waves and the role of wave‐wave interactions in limiting the wave growth. A simple mechanism, based on wave interference, is used to produce strong transient eddy growth and an analytical linear solution for the flow evolution is used as a starting point. Given the rigid upper bound on wave growth, set by the potential enstrophy conservation principle, the linear solution is bound to break down at high forcing amplitudes. An analytical quasi-linear solution, which guarantees potential enstrophy conservation, is formulated and its domain of validity is examined with a numerical nonlinear model. The nonlinear flow evolution is shown to bear strong similarity to the analytical quasi-linear solution and wave‐mean flow interactions are found to be always sufficient to limit wave growth. The saturation of the forced disturbances is shown to come through the deceleration of the mean flow and the modification of the topographic vorticity forcing. Overall, wave‐wave interactions prove not to be important in the saturation process in the examples considered. While the authors consider the implications of this result for the observationally more relevant case of vertically propagating Rossby waves, explicit calculations are clearly called for.

EXISTENCE OF PERIODIC SOLUTIONS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH DISCONTINUITIES

Article EXISTENCE OF PERIODIC SOLUTIONS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH DISCONTINUITIES was published on October 1, 2000 in the journal Demonstratio Mathematica (volume 33, issue 4).

Nonlinear propagation of Bessel–Gauss ultrasonic beams

We investigate theoretically the properties of the fundamental and second harmonic components of the Bessel beam with a finite aperture, the Bessel–Gauss beam, whose transverse profile is given by the product of Bessel and Gaussian functions. The analysis is based on the linearized and quasilinear solutions of the Khokhlov–Zabolotskaya–Kuznetsov nonlinear wave equation. The analytical and approximate expressions are derived for the fundamental and the second harmonic generation in this beam. It is thereby demonstrated that under certain circumstances, the second harmonic in the Bessel–Gauss beam is nearly radially nondiffracting, and that the beamwidth is approximately one-half of that of the fundamental. This result is an extension to the previous work on the nonlinearity of the Bessel beam, where the infinite extent of the beam has been assumed.

Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods

Shimmy oscillations are still a problem in design and operation of aircraft landing gears, and accurate and appropriate analysis is required to master the task. Based on a nonlinear model of the mechanics of the landing gear and tire elasticity according to elastic string theory, some well known linear and nonlinear mathematical methods are applied to the shimmy analysis of a simple model of a nose gear: Computing eigenvalues, solving analytically the stability boundaries with a parameter space method, getting limit cycles by analytical formulae from describing functions, and last but not least numerical simulation of time histories. It seems that linear or quasi linear methods and analytical solutions are well suited to obtain extensive insight, respecting limitations of these methods. Numerical simulation on the other hand is a valuable tool for pointing out specific effects of a nonlinear system in large amplitude regions.

A refined nonlinear analysis of pre-twisted composite blades

A nonlinear formulation for the structural behavior of initially twisted solid and thin walled composite blades is presented. The model is designed to handle arbitrary thick solid cross-sections or general thin-walled geometries, and includes three-dimensional out-of-plane warping. The nonlinear scheme enables the inclusion of geometrical nonlinearities in the presence of large initial twist, and is based on a formulation that preserves three-dimensional effects and continuity of displacements, rotations and detailed warping distributions. A basic and important feature of the present analysis is the fact that the quasi-linear solutions that are employed throughout the iterative process are executed for the blade as a whole. This allows one to determine and preserve complex three-dimensional effects including warping continuity which are usually lost by standard segment discretization schemes. A study of the combined effect of initial twist and composite induced elastic couplings is presented in addition to a correlation with nonlinear experimental data.

Application of the integrated safety assessment methodology to the protection of electric systems

The generalization of classical techniques for risk assessment incorporating dynamic effects is the main objective of the Integrated Safety Assessment Methodology, as practical implementation of Protection Theory. Transient stability, contingency analysis and protection setpoint verification in electric power systems are particularly appropriate domains of application, since the coupling of reliability and dynamic analysis in the protection assessment process is being increasingly demanded. Suitable techniques for dynamic simulation of sequences of switching events in power systems are derived from the use of quasi-linear equation solution algorithms. The application of the methodology, step by step, is illustrated in a simple but representative example.

Analysis of Water and Solute Transport Away from a Surface Point Source

AbstractMathematical theories describing the movement of water and solute away from sources of limited spatial extent mostly assume some form of three‐dimensional flow domain, but few have been rigorously tested, mainly because of the lack of experimental data. The objectives of this study were to obtain water flow and solute transport measurements during three‐dimensional, axially symmetric water flow, using time domain reflectometry (TDR), and to test some of the existing theories of three‐dimensional flow and transport using the measured data. A sandy soil was packed into a Plexiglas box fitted with curved TDR probes at different radial distances from one corner. A constant flux of water was applied at the corner of the box, and a pulse of KCl added as a tracer. Measurements of volumetric water content, θ, and resident solute concentration, CR, as a function of time and radial distance were made along four flowlines. The utility of a quasilinear solution was evaluated by comparing measured and predicted profiles of θ at steady state, and the distribution of solute travel times along various transects. Along the various streamlines, nonlinear least‐squares analysis of measured solute travel times (t*) produced estimates of the slope (α) of the exponential hydraulic conductivity function that were comparable to independently determined values. Estimation of α from t* along a single flowline, using a quasilinear solution, offers potential for field application. This technique could be used as a first step in the investigation of contaminant migration from point sources, or used in trickle‐irrigation design.

Diffraction effects in nonlinear Rayleigh wave beams

Nonlinear effects in Rayleigh wave beams are investigated analytically and numerically. Diffraction effects are taken into account within the parabolic approximation, and nonlinearity is taken into account with a theoretical model used previously to investigate plane and cylindrical Rayleigh waves of finite amplitude [E. A. Zabolotskaya, J. Acoust. Soc. Am. 91, 2569–2575 (1992)]. Quasilinear solutions are presented for second-harmonic generation in Gaussian beams, both unfocused and focused. Asymptotic results are derived for the second harmonic in the far field of arbitrary directive sources. Numerical solutions are used to investigate harmonic generation, waveform distortion, and shock formation in Rayleigh wave beams produced by sources with uniform amplitude distributions. Brief consideration is given to effects of source asymmetry on the radiated field.

Analytical method for describing the paraxial region of finite amplitude sound beams

The paraxial region of finite amplitude sound beams in lossless fluids is studied theoretically. Both focused and unfocused beams are considered. A special analytical method which combines the parabolic approximation (KZ equation) with nonlinear geometrical acoustics (NGA) is developed to model nonlinear and diffraction effects near the axis of the beam. The corresponding system of nonlinear equations describing waveform evolution is derived. For the case of an initially sinusoidal wave radiated by a Gaussian source, an analytical solution of the coupled equations is obtained along the axis of the beam. The solution is expressed in both the time and frequency domains. In the high-frequency limit, classical simple wave solutions are recovered (plane-wave solution for unfocused beams and spherical wave solution for focused beams). In the limit of weak nonlinearity, the quasilinear axial solutions of the KZ equation for the fundamental and second-harmonic components are recovered. The analytical solution is in good agreement with numerical solutions of the KZ equation for a wide range of ratios between the focal length, Rayleigh distance, and shock formation distance. Harmonic propagation curves, waveform distortion, focusing amplification factors, and other characteristics are calculated. [Work supported in part by NATO and ISF.]