Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

A criterion for the non-existence or non-uniqueness of solutions of self-similar problems in the mechanics of a continuous medium

Systems of hyperbolic partial differential equations expressing conservation laws are considered. A sufficient condition is formulated under which the self-similar problem of the disintegration of an arbitrary discontinuity (or the “piston” problem) either has no solution or the solution is not unique. This sufficient condition is determined by the existence of non-evolutionary discontinuities which may be considered as a sequence of two evolutionary discontinuities moving at the same velocity, if such a representation is unique. The condition is more general than that formulated previously, which was based on the existence of a non-proper Jouguet point. The new criterion is satisfied by weak quasitranverse shock waves in elastic media, whatever the sign of the coefficient of the non-linear deformation term. It also enables one to draw conclusions as to the non-existence or non-uniqueness of solutions of problems of the theory of elasticity in the case of finite-amplitude waves.     

Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.     

Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation

Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can describe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipation and dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.     

The decay of an arbitrary initial discontinuity in an elastic medium

The solution of the problem of the decay of an arbitrary discontinuity in elastic theory is studied. It is assumed that a plane boundary separates an elastic homogeneous, non-heat-conducting medium into two half-spaces with different elastic properties and densities. Each of the media possesses an arbitrary kind of homogneous initial strain (stress) and velocity. In the sequel the stresses and velocities of the media are assumed to be continuous at the boundary. This results in the formation of a system of plane selfsimilar waves (simple and shock), which propagate in each of the half-spaces. The problem is solved under the assumption of weak non-linearity and anisotropy of the materials. This permits an approximate evaluation of the stress and strain at the contact discontinuity. After this the problem on the decay of an arbitrary initial discontinuity is reduced to two problems on the sudden change of load on a half-space boundary, which are solved independently for each of the media.     

Special discontinuities in nonlinearly elastic media

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

The method of successive approximations for solving the problem of the decay of a discontinuity and its application to the equations of two-layer shallow water theory

The method of successive approximations for solving the problem of the decay of a small amplitude discontinuity is proposed for hyperbolic systems of conservation laws. In the linear approximation, a Cauchy problem for a linear hyperbolic system is obtained. Its solution represents lines of discontinuity separating the regions in which the solution is constant. Most attention is paid to the first and second approximations, within the limits of which the discontinuities obtained in the first approximation decay into stable shock waves and rarefaction waves. An analysis of the qualitatively different flow conditions that arise when solving the problem of the failure of a dam for a two-layer shallow water model with a free boundary is presented as an example.     

An analytical method for solving the problem of unshocked conical gas compression

The two-dimensional unsteady self-similar problem of unlimited unshocked conical compression of a gas is investigated. A solution is constructed in the form of a characteristic series in the domain bounded by a weak discontinuity and the sonic perturbation front. A recursion system of ordinary differential equations is obtained for the coefficients. A boundary-value problem corresponding to the next approximation is investigated in detail, a fundamental system of solutions is found by analytical methods and its asymptotic behaviour is investigated. Essentially independent solutions are determined and different methods are used to seek a solution of the inhomogeneous equation with the required asymptotic behaviour. An algorithm is constructed to compute gas flows induced by the motion of a piston taking the first terms of the series into consideration. The results are compared with those of computations carried out using the method of characteristics.     

Solitary waves and the structures of discontinuities in non-dissipative models with complex dispersion

Stationary solutions of reversible evolutionary equations of mechanics with higher derivatives are analysed. A two-dimensional graphical method for investigating the solutions of systems of ordinary differential equations is described, which enables one to find special types of solutions: periodic waves, solitary waves and the structures of discontinuities. At the same time, solitary waves can be obtained by taking the limit of sequences of periodic waves and the structures of discontinuities obtained by taking the limit of sequences of solitary waves. This general approach has enabled the existence of all earlier predicted structures to be verified has enabled new types of structures (three-wave structures) to be revealed and has enabled all the necessary conditions at the discontinuities to be found. All the previously known types of solitary waves are found and new types of solitary waves are revealed (generalized ordinary and 1:1 multisolitons). Methods of finding generalized solitary waves, including those with a finite amplitude of the periodic component, are determined. Examples of the solution of the following problems are given for a fourth-order system: generalized solitary waves as the limiting solutions of two-wave resonance solutions, generalized solitary waves and the structure of a discontinuity with three waves, a 1:1 soliton and the structure of a discontinuity with a single radiated wave, a solitary wave with fixed propagation velocity, and the structure of a discontinuity in the form of a kink with radiation. A generalized 1:1 soliton and the structure of a discontinuity with two radiated waves is considered in the case of sixth-order systems. The discussion is mainly based on the example of travelling waves described by the generalized Korteweg-de Vries equations. Other models with complex dispersion (a plasma and a stratified fluid) are also considered.     

一类KdV-Burgers方程的奇摄动解与孤子解

讨论了一类具有大Reynolds数且弱频散性的KdV-Burgers方程,在数学上表示为一类奇摄动KdV-Burgers方程.KdV-Burgers方程中含有的非线性项与频散项互补作用形成稳定向前传播的孤立子.通过数学分析,描述了孤立子的传播途径和传播速度等物理量的发展变化规律.通过奇摄动展开方法,构造了该问题的渐近解...     

Running waves and dynamical chaos in active media: A numerical study

In the framework of a self-similar problem, we numerically analyze the onset of solutions in the form of running waves in excitable and oscillating active media. We show that the passage to the solution in the form of a running wave occurs through cascades of bifurcations responsible for the development of chaotic dynamics in nonlinear dissipative systems of differential equations.     

Dynamically adapted grids for interacting discontinuous solutions

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is considered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approximation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuniform grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adaptation method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid consisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the diffusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking.     

Two-Parameter Asymptotics in a Bisingular Cauchy Problem for a Parabolic Equation

The Cauchy problem for a quasilinear parabolic equation with a small parameter ε at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable x/ρ, where ρ is another small parameter. This problem statement is of interest for applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters ε and ρ independently tending to zero. It is assumed that ε/ρ → 0. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of ε and ρ. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio ρ/ε. The coefficients of the inner expansion are determined from a recursive chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the internal space variable is determined.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

Self-similar solutions describing unsteady thermocapillary fluid flows

Unsteady thermocapillary flows in thin layers and layers of infinite thickness with non-uniform heating of the free boundary are investigated at high Marangoni numbers. In the plane and axially symmetric cases, self-similar solutions of the non-linear boundary-layer equations are constructed and asymptotic formulae are presented. It is shown that the self-similar solutions may be non-unique for certain values of the parameters of the problem. The branching points are calculated numerically and the branched solutions are investigated.     

Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation

In shock loading computations based on an implicit finite-difference scheme, the surfaces of velocity discontinuity and the discontinuity sizes are determined by computing an asymptotic (ray) expansion of the solution behind the front surfaces at every time step. The method for constructing ray expansions is based on a recurrence formulation of the geometric and kinematic consistency conditions for discontinuities of the derivatives of functions that are discontinuous on a moving surface. The algorithm is illustrated by computing a simple example of the out-of-plane motion of an incompressible elastic medium.     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

A solution for transient surface waves of the B-G type in a dissipative piezoelectric crystal

We derive a solution to the problem of shear horizontal electroacoustic surface waves in a piezoelectric half-space. We formulate a time-domain dynamic problem accounting for time dispersion for both electric and elastic fields and use a separation of variables to express the solution in terms of a wave propagator. Transient surface waves of the B-G type are found to propagate with a constant speed and exponentially decay in space. Their amplitude vanishes at large distances from the boundary as the reciprocal of the depth. Dispersive and non dispersive solutions are compared.