A pair of exactly soluble non-linear oscillators

A pair of anharmonic oscillators which are related to the simple pendulum are treated analytically. Decomposing the potential function of the simple pendulum into two parts, a pair of non-linear oscillators emerge. Potential functions of these non-linear oscillators are rather complicated, but complementary to each other. It is shown that the equations of these non-linear oscillators can be solved exactly using Jacobian elliptic functions. In connection with these pendulum-like oscillators, a pair of non-linear wave equations are considered and simple solutions of these wave equations are also discussed briefly.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.     

On the stability of plane wave and soliton solutions to the Zakharov equations

Plane wave and soliton solutions of the two types of Zakharov equation (two dimensional and simplified one directional) are considered. Stability properties in one dimensional space are seen to be similar. This is interesting, as the first type of equation is not solvable whereas the second is. The soliton solutions of both are one dimensionally stable but those of the full Zakharov equations are unstable with respect to perpendicular perturbations. Regions of stability of nonlinear wave and shock wave solutions in parameter space as well as growth rates of instabilities are given.     

Similarity solutions to some non-linear impact problems

Impact and wave propagation problems are considered for nonlinearly viscous and nonlinearly elastic materials. The governing partial differential equations are reduced to ordinary differential equations by means of similarity transformations. The resulting non-linear two point boundary value problems are then, in general, integrated numerically, although some closed form solutions are presented.     

Discontinuity formation in solutions of homogeneous non-linear hyperbolic equations possessing smooth initial data

Large classes of non-linear equations, at which previous breakdown theories have been aimed, are obtainable by differentiation of first order equations. The general solutions of these first order equations are also solutions of the corresponding second order equations. These are displayed and employed to calculate the critical time for singularity occurrence. Examples are discussed from gas dynamics, shallow water waves, wave propagation in solids, and electrical transmission lines. This method, when applicable, is simple and yields results which agree with those obtainable from the Ludford and Lax-Jeffrey theories.     

On the resonant nonlinear traveling waves in a thin rotating ring

Large amplitude, traveling wave motion of an inextensible, linearly elastic, rotating ring is analyzed. Equations governing the planar dynamics of a thin rod, curved in its undeformed state and moving in a horizontal reference frame which rotates about a fixed axis, are obtained via Hamilton's extended principle. The equations are specialized to study the behavior of a rotating circular ring and approximate solutions are obtained near resonance utilizing a perturbation analysis. Undamped free and viscously damped forced traveling wave motion is considered. The motion is found to consist of a forward and a backward traveling wave which may be coupled due to the non-linear terms present in the equations of motion     

Solitary waves in initially stressed thin elastic tubes

In the present work, we study the propagation of non-linear waves in an initially stressed thin elastic tube filled with an inviscid fluid. Considering the physiological conditions of the arteries, in the analysis, the tube is assumed to be subjected to a uniform inner pressure P₀ and an axial stretch ratio λ_z. It is assumed that due to blood flow, a finite dynamical displacement field is superimposed on this static field and, then, the non-linear governing equations of the elastic tube are obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the longwave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. It is observed that the present model equations give two solitary wave solutions. The results are also discussed for some elastic materials existing in the literature.     

The non-linear,dynamic response of an impulsively loaded circular plate with a central hole

A numerical method, called Direct Analysis, is described and applied to solve the problem of a plate undergoing a large impulsive load. For generality, an expanded, non-linear form of the equations of motion is used and shear correction and rotatory inertia are considered. The wave speeds are calculated from the non-linear equations and appropriate boundary conditions are applied so that reflected waves are included. The results for two types of step loading pulses are presented and compared with previously presented solutions. The response of the plate is discussed and conclusions as to the effects of the non-linearities are given.     

On time-dependent,two-layer flow over topography: II. Evolution and propagation of solitary waves

The evolution of topographically generated interfacial motion is considered in a two-layer model. A system of two non-linear equations, similar to the Boussinesq equations for shallow water waves, is derived. The consequences of the cubic non-linearity of these equations on the nature of the solitary wave solutions are explored. A dispersion relation for solitary waves implies the existence of maxima for speed and displacement in a wave. The limiting values are shown to agree with other studies. The growth of solitary and/or cnoidal waves is studied for finite pulses of displacement and for internal bores.     

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium are investigated. The entire temperature range is divided into a number of small sub-regions where the thermal properties can be approximated to be constant. The resulting problems can be considered as the Stefan’s problem of a multi-phase with no latent heat and the exact solutions called Neumann’s solution are available. In order to obtain the solutions, however, a set of highly non-linear equations in determining the phase boundaries should be solved simultaneously. This work presents a semi-analytic algorithm to determine the phase boundaries without solving the highly non-linear equations. Results show that the solutions for a set of highly non-linear equations depend strongly on the initial guess, bad initial guess leads to the wrong solutions. However, the present algorithm does not require the initial guess and always converges to the correct solutions.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.     

Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler–Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator–Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.     

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

Complete group classification and exact solutions to the extended short pulse equation

In this paper, the complete group classification is performed on the extended short pulse equation (ESPE), which including many important non-linear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary parameters of the equation. Furthermore, the symmetry reductions and exact solutions to the short pulse types of equations are investigated, and the physical significance of the solutions are considered from the transformation group point of view.     

The method of successive integration of the linear inhomogeneous wave equations in the theory of transient wave propagation in non-linear hereditary elastic media

A moderate distortion of the initial pulse form which takes place when a one-dimensional longitudinal pulse propagates through a sufficiently small distance in a non-linear hereditary clastic medium is considered. The governing equation is a quasi-linear integro-differential equation. Its first- and second-order asymptotic solutions arc derived with the aid of a method of successive integration of the linear inhomogeneous wave equations. Besides the constants which define the wave speed and the non-linear properties of the medium, the asymptotic solutions suggested in this paper contain two arbitrary functions whose properties are restricted only by certain smoothness conditions. One of them is the kernel function which defines the hereditary properties of the medium. and the other is the function which defines the initial form (shape) of the pulse. An example of the use of the asymptotic solutions is presented in which these two functions are given explicitly.     

A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction

A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N>2 first-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of field equations induces the hyperbolicity of an auxiliary 2×2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic fluid is considered. For appropriate model pressure-entropy-density laws, we determine a solution to the governing system of equations which describes in the 2+1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with unaffected profile from the interaction region. These model material laws include the classical pressure-entropy-density law which is usually adopted for a polytropic fluid.