Extending the generalized tanh method to the generalized Hamiltonian equations: New soliton-like solutions

To certain nonlinear evolution equations, the tanh method has been generalized for constructing not only solitary-wave but also soliton-like solutions. In this paper, no loss of conciseness, we further extend the generalized tanh method with computerized symbolic computation to a pair of generalized Hamiltonian equations. A new family of soliton-like analytical solutions is obtained, of which the solitary waves and previously-claimed soliton-like solutions are shown to be the special cases.     

A note on solitary waves in viscoelastic porous media

Solitary waves occur in the system of wave propagation in a dispersive energy conserving medium. We show that soliton-like travelling waves arise naturally in the flow of viscoelastic media.     

Modelling of Lamb wave propagation in composite plate excited by surface bonded piezoelectrical actuators

Transient propagation of Lamb waves in a multilayered infinite composite plate with the arbitrary elastic anisotropy of each layer is considered in this paper. Integral representations of wave fields, an algorithm of computing the Green's matrix of the multilayered medium in Fourier domain and some methods of the Fourier transform inversion are used as basic analytical research instruments. The piezoelectrical transducers frequently used for Lamb wave excitation are modelled in this paper as surface sources with stresses concentrated on the boundaries of actuators. The contribution of each wave mode to a whole amplitude, focusing of Lamb waves and influence of the geometry of the surface source are discussed in this paper by numerical examples. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effects of initial stresses on guided wave propagation in multilayered PZT-4/PZT-5A composites: A polynomial expansion approach

Effects of initial stresses on the dispersion curves of Lamb and SH waves in multilayered PZT-4/PZT-5A composites are investigated using the polynomial expansion approach. The piezoelectric layers are considered with arbitrary crystal orientations with a result that only Lamb or SH waves may be transmitted. The problem is solved employing the Legendre polynomial approach that poses the advantages of numerically stability and effectiveness over conventional matrix method. The solution is validated by comparing the wave propagation behavior of piezoelectric materials with those reported in literature, and the convergence properties are examined. Numerical results demonstrate that initial stress has profound influences on the guided wave propagation in multilayered PZT-4/PZT-5A laminates. The phase velocity of Lamb and SH waves increases with initial tensile stresses. In addition, the effects of initial stresses rely on the wave mode and thickness of constituent layers and the stacking sequence of the constituent materials. The results are useful for understanding and optimization of new designs for actuator, electromechanical sensor and acoustic wave devices made of PZT-4/PZT-5A composites.     

Shock waves in discrete nonlinear Schrödinger equations

It is shown that in nonlinear differential–difference equations, which in the continuum limit are reduced to the nonlinear Schrödinger equation, localized excitations, reminding shock waves in liquids and gasses can propagate. Such waves may have either the conventional profile of shock waves or a shape of dark pulses evolving against a background. At the initial stages of evolution the shock waves are described by the equation u_t+uu_x=0 and split out in a train of soliton-like pulses after the shock is developed.     

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.     

Lamb Wave Interactions with Non-symmetric Features at Structural Boundaries

The paper initially describes on a numerical basis how a Lamb wave would have to perform that has been initiated in a pure mode (either symmetric or anti–symmetric) and what the wave would have to anticipate in terms of mode conversion when being reflected at a surface not perpendicular to its traveling direction. The effects of changing in geometric specifications of non–symmetric artificial features like angle of sloping edge or partially sloping edges are studied. The results obtained from these studies are presented as the reflected and converted parts of the incident wave versus angle of the edge or percentage of the sloped edge. It has been further shown that Lamb waves being generated experimentally by a finite size transducer into a plate like structure thus most likely result in a combination of modes. Reflection of these combined modes at structural boundaries will therefore generate an even more complex coupling of modes. This situation is further aggravated if the structural boundary is not purely perpendicular to the traveling wave but has a slightly varying angle such as it might have to be anticipated at a countersunk rivet, a notch or even more extreme a crack in a metallic component. However from understanding the background of Lamb wave generation, mode separation and superposition, a systematic approach can be established that allows complex Lamb waves, such as they are observed when monitoring true structures, to be interpreted and understood. This approach has been explained on the basis of numerical result obtained from finite element analyses first before proving the findings by some fundamental experiments performed with variable angle beam transducers which demonstrates the difficulties in de–coupling Lamb wave modes and how to handle those coupled modes in terms of structural condition monitoring. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

松弛时间对热弹性扩散板中环形峰波的影响

在不同热弹性扩散理论的框架中,无应力作用、恒温/绝热、化学势条件下,研究了均匀、横观各向同性无穷板中热弹性扩散环形峰波的传播．得到了热弹性扩散Lamb型波的色散方程．同时推演出色散方程的一些特例．     

Dispersion of lamb waves in a three-layer plate made from compressible materials with finite initial deformations

A three-layer plate made from compressible elastic materials with finite initial deformations caused by compressive forces applied to their faces is considered. By using the three-dimensional linearized theory of propagation of elastic waves in bodies with initial stresses, the propagation of Lamb waves in this plate is investigated.     

Solitary waves in a cold plasma

We study the existence of soliton-like solutions (solitary waves) to the equations describing the one-dimensional motion of a cold quasi-neutral plasma. It is shown that in some range of the angle between the norperturbed magnetic field and the wave propagation direction there exists a branch of solitary hydromagnetic waves that is a bifurcation of the zero wave number. These solutions lie on a two-dimensional center manifold.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 719–728, May, 1996.     

Effect of absorption in the medium on the damping of Lamb waves in a well

The effect of absorption of longitudinal, transverse, and acoustic waves in the medium and the well fluid on the damping of Lamb waves recorded during acoustic logging is investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 57–63, 1980.     

Damage detection in plates by wave propagation analyses

At locations with defects the propagation of elastic waves in solids is disturbed. This behaviour can be used for the development of new structural health monitoring and quality assurance systems. In the sense of adaptive structure systems actuators of piezo–electric materials [1] are implemented to generate high frequency waves (Lamb waves) in plate like structures, [2]. At flaws, imperfections of the structure and surface defects these waves show reflections, refractions and mode conversions. The observation of the wave propagation and the wave behaviour at possible flaws and surface defects is performed by an optical measurement system. Measurement data of the thin walled metallic plates are compared with numerical and analytical solutions. First results of the detection of the wave propagation in metallic plates are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

What is sustainable development? An attempt to interpret it as a soliton-like phenomenon

An attempt is made to establish a relation between the question of what ‘sustainable development’ means and the non-linear theory of shock waves. Despite the presence of dispersive, i.e. entropy-producing, forces a soliton-like, isentropic, transport of a wilfully desired distribution in a field of traded commodities is possible. Starting with the classical Korteweg de Vries (KdV) equation, two other examples, a sigmoidal and a Gaussian soliton in a diffusional environment, are analyzed in detail as a guide-line of how a ‘sustainable’ transport of an economically defined creation can be carried through time.     

The excitation of symmetric and antisymmetric lamb waves in a piezoelectric strip by surface electrodes

Solutions of the problems of the excitation by surface electrodes of symmetric and antisymmetric Lamb waves in a strip of piezoelectric material are obtained. The solutions of the problems are reduced to systems of singular integral equations in auxiliary functions, proportional to the jump in charge density at the electrodes. Approximate solutions of the systems of singular integral equations are obtained by the Bubnov-Galerkin method using Chebyshev polynomials of the first kind. The results of numerical calculations of the electro-elastic fields for a strip of cadmium sulphide are given.     

The asymptotic form of the spectrum of tidal Kelvin waves trapped by a boundary at large values of the lamb parameter

Tidal Laplace equations are considered and the asymptotic form of the eigenvalues corresponding to eigenfunctions (Kelvin waves) concentrated in the neighbourhood of a closed component of the boundary are constructed for large values of the Lamb parameter, which is proportional to the square of the angular velocity of rotation of the reservoir. The formulae obtained determine the asymptotic behaviour of the kth eigenvalue in the case of a planar, simply connected reservoir of constant depth.     

Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

In this paper, a generalized variable-coefficient Gardner equation arising in nonlinear lattice, plasma physics and ocean dynamics is investigated. With symbolic computation, the Lax pair and Bäcklund transformation are explicitly obtained when the coefficient functions obey the Painlevé-integrable conditions. Meanwhile, under the constraint conditions, two transformations from such an equation either to the constant-coefficient Gardner or modified Korteweg-de Vries (mKdV) equation are proposed. Via the two transformations, the investigations on the variable-coefficient Gardner equation can be based on the constant-coefficient ones. The N-soliton-like solution is presented and discussed through the figures for some sample solutions. It is shown in the discussions that the variable-coefficient Gardner equation possesses the right- and left-travelling soliton-like waves, which involve abundant temporally-inhomogeneous features.     

Reconstruction of a Source from Observational Data

A procedure is described for determining the strength, position, and startup time of a surface force source acting on an elastic half-space from recorded data on the vertical displacements (seismograms) of a linear oscillator (seismograph). The algorithm is based on a previously published analytical solution of the Lamb problem. Special attention is given to the role of the Rayleigh wavefront as a generator of displacements much larger than in the approach of P waves and SV waves to the seismograph, revealing the Rayleigh wave as the primary indicator of a source's space–time position.     

Verification and reformulation of the competitive exclusion principle

Biodiversity conservation is becoming increasingly urgent. It is important to find mechanisms of competitive coexistence of species with different fitness in especially difficult circumstances – on one limiting resource, in isolated stable uniform habitat, without any trade-offs and cooperative interactions. Here we show a such mechanism of competitive coexistence based on a soliton-like behaviour of population waves. We have modelled it by the logical axiomatic deterministic individual-based cellular automata method. Our mechanistic models of population and ecosystem dynamics are of white-box type and so they provide direct insight into mechanisms under study. The mechanism provides indefinite coexistence of two, three and four competing species. This mechanism violates the known formulations of the competitive exclusion principle. As a consequence, we propose a fully mechanistic and most stringent formulation of the principle.     

Excitation of Lamb-waves in plates

Lamb waves show great potential in the detection of structural defects in plate and shell structures. In the sense of the adaptive structures concept the wave generation takes place with help of integrated piezoelectric ceramics. For a steel plate with adhered piezoelectric actuator, analytical calculations, FEM simulations, and experimental results are in good agreement. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)