1,064 research outputs found
Discrete Analog of the Burgers Equation
We propose the set of coupled ordinary differential equations
dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers
equation. We focus on traveling waves and triangular waves, and find that these
special solutions of the discrete system capture major features of their
continuous counterpart. In particular, the propagation velocity of a traveling
wave and the shape of a triangular wave match the continuous behavior. However,
there are some subtle differences. For traveling waves, the propagating front
can be extremely sharp as it exhibits double exponential decay. For triangular
waves, there is an unexpected logarithmic shift in the location of the front.
We establish these results using asymptotic analysis, heuristic arguments, and
direct numerical integration.Comment: 6 pages, 5 figure
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
Nonlinear transient waves in coupled phase oscillators with inertia
Like the inertia of a physical body describes its tendency to resist changes
of its state of motion, inertia of an oscillator describes its tendency to
resist changes of its frequency. Here we show that finite inertia of individual
oscillators enables nonlinear phase waves in spatially extended coupled
systems. Using a discrete model of coupled phase oscillators with inertia, we
investigate these wave phenomena numerically, complemented by a continuum
approximation that permits the analytical description of the key features of
wave propagation in the long-wavelength limit. The ability to exhibit traveling
waves is a generic feature of systems with finite inertia and is independent of
the details of the coupling function.Comment: 12 pages, 4 figure
The Riccati System and a Diffusion-Type Equation
We discuss a method of constructing solution of the initial value problem for
duffusion-type equations in terms of solutions of certain Riccati and
Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.Comment: 11 pages, no figure
Transient granular shock waves and upstream motion on a staircase
A granular cluster, placed on a staircase setup, is brought into motion by vertical shaking. Molecular dynamics simulations show that the system goes through three phases. After a rapid initial breakdown of the cluster, the particle stream organizes itself in the form of a shock wave moving down the steps of the staircase. As this wave becomes diluted, it transforms into a more symmetric flow, in which the particles move not only downwards but also toward the top of the staircase. This series of events is accurately reproduced by a dynamical model in which the particle flow from step to step is modeled by a flux function. To explain the observed scaling behavior during the three stages, we study the continuum version of this model (a nonlinear partial differential equation) in three successive limiting cases. (i) The first limit gives the correct t−1/3 decay law during the rapid initial phase, (ii) the second limit reveals that the transient shock wave is of the Burgers type, with the density of the wave front decreasing as t−1/2, and (iii) the third limit shows that the eventual symmetric flow is a slow diffusive process for which the density falls off as t−1/3 again. For any finite number of compartments, the system finally reaches an equilibrium distribution with a bias toward the lower compartments. For an unbounded staircase, however, the t−1/3 decay goes on forever and the distribution becomes increasingly more symmetric as the dilution progresses
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