Travelling, non-periodic patterns in nonlinear stability problems

In this short note we present results on the existence of several classes of travelling, non-periodic solutions of the complex Ginzburg-Landau equation. First we give a very short introduction to the G-L equation and show its importance in nonlinear stability theory. We then study the G-L equation with complex coefficients and establish the existence of a 2-parameter family of quasi-periodic solutions and two different types of one-parameter families of heteroclinic orbits; all members of these families travel with a well-defined wave-speed. The heteroclinic solutions correspond to (travelling) soliton-like localized structures which connect different (stable) periodic patterns. Mathematically, these families of travelling solutions (quasi-periodic and heteroclinic) are continuations into the complex case of the stationary solutions of the real G-L equation.     

Solitons in Josephson junctions: An overview

The dynamics of the Josephson tunnel junction is approximately described by a perturbed sine-Gordon equation. The Josephson tunnel junction is thus a convenient experimental solid state device for the study of solitons and solitonlike phenomena. The physical manifestation of the soliton is a propagating magnetic flux quantum ( ₀=h/2e=2.064×10^–15 V sec). Basic properties of the soliton and its relation to observable experimental quantities (zero field steps, microwave radiation, etc.) are reviewed. Recent direct measurements of the actual soliton profile are also mentioned.     

Shock-wave equation-of-state studies at Los Alamos

A history of the shock-wave equation-of-state (EOS) studies at Los Alamos is given. Particular emphasis is placed on the pioneering research in the 1950s where many of the experimental techniques and methods of analysis were developed, which we now take for granted. A brief review of shock-wave physics is given, which illustrates important hydrodynamic and thermodynamic concepts. Recent studies on the EOS of Ti are presented with emphasis on the-to- phase transition. VISAR wave profiles for polycrystalline Ni and singlecrystal Ni are presented to determine the strengths of these materials under pressure. Low-density polystyrene foam Hugoniot experiments are described and results analyzed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Factors affecting propagating fronts of addition polymerization: Velocity,front curvature,temperatue profile,conversion, and molecular weight distribution

Several properties of propagating fronts of addition polymerization were studied. A power function could be fit to the velocity dependence on initiator concentration, but not with the exponents predicted by current models or in agreement with other published work. Bubbles from the volatile by-products of initiator decomposition were found to affect the front velocity and curvature. The front velocity for triethylene glycol dimethacrylate polymerization was found to depend linearly on temperature over a moderate range. The conversion of methacrylic acid in fronts varied greatly with initiator type and concentration. Benzoyl peroxide produced much lower conversion than t-butyl peroxide, but fronts with tBPO propagated slower. A dual initiator system of BPO and tBPO produced rapidly propagating fronts with good conversion but the contribution of each initiator to the velocity was not additive. The possibility of chain branching was considered. The apparent molecular weight distributions were very broad, often trimodal, and found to depend on initiator type and concentration as well as the tube diameter. The temperature profiles were measured and found to be very sharp for BPO and broader for tBPO but both had front temperatures in excess of 200°C, indicating a high ceiling temperature. © 1995 John Wiley & Sons, Inc.     

Sloshing waves in a heated viscoelastic fluid layer in an excited rectangular tank

In this paper, we have investigated the motion of a heated viscoelastic fluid layer in a rectangular tank that is subjected to a horizontal periodic oscillation. The mathematical model of the current problem is communicated with the linearized Navier–Stokes equation of the viscoelastic fluid and heat equation together with the boundary conditions that are solved by means of Laplace transform. Time domain solutions are consequently computed by using Durbin's numerical inverse Laplace transform scheme. Various numerical results are provided and thereby illustrated graphically to show the effects of the physical parameters on the free-surface elevation time histories and heat distribution. The numerical applications revealed that increasing the Reynolds number as well as the relaxation time parameter leads to a wider range of variation of the free-surface elevation, especially for the short time history.     

Shock waves for the hyperbolic balance laws with discontinuous sources

In this paper, we are concerned with the behavior of shock waves in a 2 × 2 balance law with discontinuous source terms. We obtain the existence of a local shock wave solution of this problem and deduce that the discontinuous source terms create a weak discontinuity in this solution. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ²ⁿ⁺²) term (n ∈ ?) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P₂ Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Anisotropic sound and shock waves in dipolar Bose-Einstein condensate

We study the propagation of anisotropic sound and shock waves in dipolar Bose-Einstein condensate in three dimensions (3D) as well as in quasi-two (2D, disk shape) and quasi-one (1D, cigar shape) dimensions using the mean-field approach. In 3D, the propagation of sound and shock waves are distinct in directions parallel and perpendicular to dipole axis with the appearance of instability above a critical value corresponding to attraction. Similar instability appears in 1D and not in 2D. The numerical anisotropic Mach angle agrees with theoretical prediction. The numerical sound velocity in all cases agrees with that calculated from Bogoliubov theory. A movie of the anisotropic wave propagation in a dipolar condensate is made available as supplementary material.