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Memory effects in transport require, for their incorporation into reaction-diffusion investigations, a generalization of traditional equations. The well-known Fisher's equation, which combines diffusion with a logistic nonlinearity, is generalized to include memory effects, and traveling wave solutions of the equation are found. Comparison is made with alternative generalization procedures.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.
A phenomenological model of parametric surface waves (Faraday waves) is introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large-scale flows (mean flow). The primary bifurcation of the model is to a set of standing waves (stripes, given the functional form of the model nonlinearities chosen here). Our results for the secondary instabilities of the primary wave show that the mean flow leads to a weak destabilization of the base state against Eckhaus and transverse amplitude modulation instabilities, and introduces a longitudinal oscillatory instability which is absent without the coupling. We compare our results with recent one-dimensional amplitude equations for this system systematically derived from the governing hydrodynamic equations.
The electric field computed by numerically solving the one-dimensional Vlasov-Poisson system is used to calculate Lagrangian trajectories of particles in the wave-particle resonance region. The analysis of these trajectories shows that, when the initial amplitude of the electric field is above some threshold, two populations of particles are present: a first one located near the separatrix, which performs flights in the phase space and whose trajectories become ergodic and chaotic, and a second population of trapped particles, which displays a nonergodic dynamics. The complex, nonlinear interaction between these populations determines the oscillating long-time behavior of solutions.
The classical stimulated Raman scattering system describing the resonant interaction between two electromagnetic waves and a fast relaxing medium wave is studied by introducing a systematic perturbation approach in powers of the relaxation time. We separate amplitude and phase effects for these complex fields. The analysis of the former shows the existence of a stagnation distance after which monotonic energy transfer begins from one electromagnetic wave to the other, and this quantity is calculated. Concerning phase effects we give the conditions for the formation of a Raman spike from an initial fast and large phase jump in one of the waves. The spike evolution and width estimated from the reduced model agree with the results from numerical simulations of the original system.
Relativistic Bernstein modes are not totally undamped, but have a small, negative definite imaginary frequency component that peaks where the frequency is closest to the rest cyclotron harmonic.
The effects of radiation reaction on electron beam dynamics are studied in the context of plasma-based accelerators. Electrons accelerated in a plasma channel undergo transverse betatron oscillations due to strong focusing forces. These oscillations lead to emission by the electrons of synchrotron radiation, with a corresponding energy loss that affects the beam properties. An analytical model for the single particle orbits and beam moments including the classical radiation reaction force is derived and compared to the results of a particle transport code. Since the betatron amplitude depends on the initial transverse position of the electron, the resulting radiation can increase the relative energy spread of the beam to significant levels (e.g., several percent). This effect can be diminished by matching the beam into the channel, which could require micron sized beam radii for typical values of the beam emittance and plasma density.
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