Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.     

Generation of axisymmetric magnetosonic shock waves: Applications to static equilibria and accretion disks

Magnetosonic waves traveling in a MHD equilibrium may evolve into shocks. We develop a criterion for the creation of fast shocks in the equatorial plane of axisymmetric equilibria and analyze the influence of the most important parameters. The results are applied to Grad–Shafranov equilibria and accretion disks.     

Traveling Waves in Discrete Periodic Media for Bistable Dynamics

This paper is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for a system of ordinary differential equations exhibiting bistable dynamics. The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics. The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media.     

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.     

Pointwise Estimates and Stability for Dispersive–Diffusive Shock Waves

We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks. We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L ^p norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems. Accepted May 29, 2000?Published online November 16, 2000     

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar–Gross–Krook (BGK) numerical model for one‐ and two‐dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation‐free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one‐ and two‐dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C_r), with the only condition being C_r≤1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks). In addition, comparisons of numerical tests with measured data from dam break laboratory experiments show good agreement for C_r≤0.6. This reduction in the stability domain is due to the explicit integration of the friction term. Furthermore, BGK schemes are easily extended to multidimensional problems and do not require characteristic decomposition. The proposed scheme is second‐order in both space and time when the external forces are zero and second‐order in space but first‐order in time when the external forces are non‐zero. However, since all the test cases presented are either for zero or small values of external forces, the results tend to maintain second‐order accuracy. In problems where the external forces become significant, it is possible to improve the order of accuracy of the scheme in time by, for example, applying the Runge–Kutta method in the integration of the external forces. Copyright © 2001 John Wiley & Sons, Ltd.     

A Topological Proof of Stability of N-Front Solutions of the FitzHugh–Nagumo Equations

Consideration is devoted to traveling N-front wave solutions of the FitzHugh–Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.     

Combustion Fronts in a Porous Medium with Two Layers

We study a model for the lateral propagation of a combustion front through a porous medium with two parallel layers having different properties. The reaction involves oxygen and a solid fuel. In each layer, the model consists of a nonlinear reaction–diffusion–convection system, derived from balance equations and Darcy’s law. Under an incompressibility assumption, we obtain a simple model whose variables are temperature and unburned fuel concentration in each layer. The model includes heat transfer between the layers. We find a family of traveling wave solutions, depending on the heat transfer coefficient and other system parameters, that connect a burned state behind the combustion front to an unburned state ahead of it. These traveling waves are strong: they correspond to connecting orbits of a system of five ordinary differential equations that lie in the unstable manifold of a hyperbolic saddle and the stable manifold of a nonhyperbolic equilibrium. We argue that for physically relevant initial conditions, traveling waves that correspond to connecting orbits that approach the nonhyperbolic equilibrium along its center direction do not occur. When the heat transfer coefficient is small, we prove that strong traveling waves exist for a small range of system parameters, near parameter values where the two layers individually admit strong traveling waves with the same speed. When the heat transfer coefficient is large, we prove that strong traveling waves exist for a very large range of parameters. For small heat transfer, combustion typically does not occur simultaneously in the two layers; for large heat transfer, it does. The proofs use geometric singular perturbation theory. We give a numerical method to solve the nonlinear problem, and we present numerical simulations that indicate that the traveling waves we have found are in fact the dominant feature of solutions.     

Nonlinear Stability of Time-Periodic Viscous Shocks

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green’s distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

Optimization of a shock-wave system

The problem of a system consisting of n+m shock waves which realizes the maximum dynamic pressure is solved for given Mach numbers ahead of the first and the closing shocks provided that the sum of the flow turning angles in the last m waves is equal to the sum of the turning angles in the initial n waves minus the angle of attack. The corresponding necessary conditions of optimality of this shock-wave system, which constitutes a system of nonlinear algebraic equations, are obtained. An efficient iteration method of solving this system of equations, which makes it possible to solve the problem with high accuracy, is developed. The results of solving the problem make it possible to select the optimum supersonic air intake configuration in the preliminary design stage and in the case of a large number of shocks to estimate the limiting air intake parameters and the processes taking place in the air intake.     

Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S _{i, j} dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S _{i, j} shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S _{i, j} shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S _{i, j} shocks and classical waves are described. As a physical application, all types of S _{i, j} shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.     

Computational analysis of dense gas shock tube flow

Nonclassical phenomena associated with the classical dynamics of real gases in a conventional shock tube are studied. A TVD predictor-corrector (TVD-MacCormack) scheme with reflective endwall boundary conditions is used for the one-dimensional Euler equations to simulate the evolution of the wave field of a van der Waals gas. Depending upon the initial conditions of the gas, wave fields are produced that contain nonclassical phenomena such as expansion shocks, composite waves, splitting shocks, etc. In addition, the interactions of waves reflected from the endwalls produce both classical and nonclassical phenomena. Wave field evolution is depicted using plots of the flow variables at specific times and withx-t diagrams.     

Determination of a system consisting of 2<Emphasis Type="Italic">n</Emphasis> shock pairs realizing the total pressure maximum

The problem of a symmetric system consisting of 2n pairs of intersecting shock waves in a plane breaking duct which realizes the maximum total pressure is solved for given Mach numbers upstream of the leading shocks and downstream of the closing shocks provided that in each pair consisting of impinging and reflected waves the flow turning angles are equal in absolute values and have opposite directions. The corresponding necessary conditions of optimality of this shock-wave system, which constitutes a system of nonlinear algebraic equations, are obtained. An efficient iteration method of solving this system of equations, which makes it possible to solve the above-mentioned problem with high accuracy, is developed. An approximate analytic solution is obtained for large n. The results of solving the problem make it possible to select the optimum configuration of the plane internal-compression duct.     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

Nonlinear forced vibrations of a cylindrical shell with two internal resonances

Forced vibrations of cylindrical shells described by a system of three ordinary differential equations are studied. There are two internal resonances. Standing and traveling waves in the shells are described by a system of six modulation equations derived using the multiple-scales method. These waves are analyzed for stability __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 51–58, February 2006.     

Stability of Concatenated Traveling Waves

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is individually asymptotically stable. We show that the sequence of traveling waves is itself asymptotically stable: as \(t\rightarrow \infty \), the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Essentially the same result was previously proved by Wright (J Dyn Differ Equ 21:315–328, 2009) and Selle (Decomposition and stability of multifronts and multipulses, 2009), who regarded the concatenated wave pattern as a sum of traveling waves. In contrast to their work, we regard the pattern as a sequence of traveling waves restricted to subintervals of \(\mathbb {R}\) and separated at any finite time by small jump discontinuities. Our proof uses spatial dynamics and Laplace transform.     

Shock wave propagation and admissibility criteria in a nonlinear dielectric medium

We study the propagation of electromagnetic shock waves in an isotropic nonlinear dielectric medium. In order to select the physical shocks among all the mathematical solutions the usualLax conditions are applied. However, here they do not appear sufficient since strong shocks are present and the differential system is not strictly hyperbolic. So, two additional criteria are studied, theentropy growth condition and thereflection and transmission criterion, and a comparative analysis is developed. Finally, some experimental checks are suggested considering in particular the possible shape changes of an initial shock wave during its propagation.     

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.     

Stability of wave regimes in film flow along a vertical surface

The stability of nonlinear traveling waves with respect to all possible two-dimensional infinitesimal perturbations is numerically investigated. The stability zones are determined for two families. It is shown that regimes of the second family, which in the limit go over into positive solitons, are the more stable. Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–131, September–October, 1988.