Amplitude Oscillations of a Monochromatic Wave in a One-Dimensional Plasma

The amplitude oscillations of a monochromatic wave propagating in a one-dimensional Vlasov plasma are studied numerically. The results for the nonlinear evolution of the wave show very good qualitative agreement with the theory.     

On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network

We investigate existence and uniqueness of solutions of a McKean–Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh–Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.     

Total linearization of probability density evolution equations

Total linearization of the Boltzmann and Vlasov equations is used to present a technique applicable to equations which have polynomial-type nonlinearities and describe the evolution of probability density functions or distributions. It is an extension of an earlier result, where a method yielding a linear equation with a nonlinear constraint was presented.     

Multi-moment advection scheme for Vlasov simulations

We present a new numerical scheme for solving the advection equation and its application to Vlasov simulations. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, for better conservation of the information entropy. We have developed one-and two-dimensional schemes and show that they provide quite accurate solutions within reasonable usage of computational resources compared to other existing schemes. The two-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile for more than hundred rotation periods with little numerical diffusion. This is crucially important for Vlasov simulations of magnetized plasmas. Applications of the one- and two-dimensional schemes to electrostatic and electromagnetic Vlasov simulations are presented with some benchmark tests.     

Consequences of coarse-grained Vlasov equations

The Vlasov equation is analyzed for coarse-grained distributions resembling a finite width of test particles as used in numerical implementations. It is shown that this coarse-grained distribution obeys a kinetic equation similar to the Vlasov equation, but with additional terms. These terms give rise to entropy production indicating dissipative features due to a nonlinear mode coupling. The interchange of coarse graining and dynamical evolution is discussed with the help of an exactly solvable model for the self-consistent Vlasov equation and practical consequences are worked out. By calculating analytically the stationary solution of a general Vlasov equation we can show that a sum of modified Boltzmann-like distributions is approached dependent on the initial distribution. This behavior is independent of degeneracy and only controlled by the width of test particles. The condition for approaching a stationary solution is derived and it is found that the coarse graining energy given by the momentum width of test particles should be smaller than a quarter of the kinetic energy. Observable consequences of this coarse graining are: (i) spatial correlations in observables, (ii) too large radii of clusters or nuclei in self-consistent Thomas–Fermi treatments, (iii) a structure term in the response function resembling vertex correction correlations or internal structure effects and (iv) a modified centroid energy and higher damping width of collective modes.     

A discontinuous Galerkin method for the Vlasov–Poisson system

A discontinuous Galerkin method for approximating the Vlasov–Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates of the approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov–Poisson system.     

Localization of intense electromagnetic waves in a relativistically hot plasma

We consider nonlinear interactions between intense short electromagnetic waves (EMWs) and a relativistically hot electron plasma that supports relativistic electron holes (REHs). It is shown that such EMW-REH interactions are governed by a coupled nonlinear system of equations composed of a nonlinear Schro dinger equation describing the dynamics of the EMWs and the Poisson-relativistic Vlasov system describing the dynamics of driven REHs. The present nonlinear system of equations admits both a linearly trapped discrete number of eigenmodes of the EMWs in a quasistationary REH and a modification of the REH by large-amplitude trapped EMWs. Computer simulations of the relativistic Vlasov and Maxwell-Poisson system of equations show complex interactions between REHs loaded with localized EMWs.     

Short-time correlations of many-body systems described by nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations

Analytical expressions for short-time correlation functions, diffusion coefficients, mean square displacement, and second order statistics of many-body systems are derived using a mean field approach in terms of nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations. The results are illustrated for the Desai–Zwanzig model, the nonlinear diffusion equation related to the Tsallis statistics, and a Vlasov–Fokker–Planck equation describing bunch particles in particle accelerator storage rings.     

Vlasov moments, integrable systems and singular solutions

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie-Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.     

Study of propagation of ultraintense electromagnetic wave throughplasma using semi-Lagrangian Vlasov codes

Interaction of relativistically strong laser pulses with underdense and overdense plasmas is investigated by a semi-Lagrangian Vlasov code. These Vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by the electromagnetic wave as electron trapping, particle acceleration, and electron plasma wavebreaking. To describe the distribution of accelerated particle momenta and energy will require a very detailed analysis of the kinetic and time history of the plasma wave evolution. The semi-Lagrangian Vlasov code allows us to handle the interaction of ultrashort electromagnetic pulse with plasma at strongly relativistic intensities with a great deal of resolution in phase space     

Numerical Simulation of Plasma Edge Turbulence Due to E×B Flow Velocity Shear

A numerical code is used to study the nonlinear evolution of the Kelvin-Helmholtz instability, and the existence and time evolution of a charge separation with the self-consistent electric field at a plasma edge. Both ions and electrons are described by gyrokinetic equations that include the ions finite Larmor radius correction and the polarization drift. We present results for the case where the plasma layer is two-dimensional, and the magnetic field makes an angle very close to the normal to the plane of the plasma. At = 88.5°, the nonlinear evolution of the Kelvin-Helmholtz instability shows a spectrum which is turbulent and is dominated by higher harmonics, saturates at low level and has little effect on the electrons and ions initial equilibrium density profiles. At an angle of the magnetic field closer to 90°, when the component of the motion of the electrons along the magnetic field decreases, the behavior is in accordance with some basic physics associated with the set of equations describing the behavior of a guiding center plasma in a strong magnetic field, namely the energy condensing in the lowest k modes (inverse cascades). The results show the sensitivity of the turbulent spectrum to the motion of the electrons along magnetic field lines. In particular, we study the effect of different algorithms to compute this sensitive electrons motion, and their effect on the turbulent spectrum. We show that Eulerian Vlasov codes associated with cubic spline interpolations perform favorably when compared to other methods.     

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.     

Lateral phase drift of the topological charge density in stochastic optical fields

The statistical distributions of optical vortices or topological charge in stochastic optical fields can be inhomogeneous in both transverse directions. Such two-dimensional inhomogeneous vortex or topological charge distributions evolve in a complex way during free-space propagation. While the evolution of one-dimensional topological charge densities can be described by a linear diffusion process, the evolution of two-dimensional topological charge densities exhibits some additional nonlinear dynamics. Here we propose a phase drift mechanism as a partial explanation for this additional nonlinear dynamics. Numerical results are presented in support of this proposal.     

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schr?dinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.     

Vlasov equation on a symplectic leaf

The infinite dimensional phase space of the Vlasov equation is foliated by symplectic manifolds (leaves) which are invariant under the dynamics. By adopting a Lie transform representation, exp{W, }, for near-identity canonical transformations we obtain a local coordinate system on a leaf. The evolution equation defined by restricting the Vlasov equation to the leaf is approximately represented by the evolution of W. We derive the equation for ∂_tW and show that it is hamiltonian relative to the nondegenerate Kirillov-Kostant-Souriau symplectic structure.     

Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy

We study many-body interactions between excitons in semiconductors by applying the powerful technique of optical two-dimensional Fourier transform spectroscopy. A two-dimensional spectrum correlates the phase (frequency) evolution of the nonlinear polarization field during the initial evolution and the final detection period. A single two-dimensional spectrum can identify couplings between resonances, separate quantum mechanical pathways, and distinguish among microscopic many-body interactions.     

Hamiltonian systems with indefinite kinetic energy

Some interesting features of a class of two-dimensional Hamiltonians with indefinite kinetic energy are considered. It is shown that such Hamiltonians cannot be reduced, in general, to an equivalent dynamical Hamiltonian with positive definite kinetic energy quadratic in velocities. Complex nonlinear evolution equations like the K-dV, the MK-dV and the sine-Gordon equations possess such Hamiltonians. The case of complex K-dV equation has been considered in detail to demonstrate the generic features. The two-dimensional real systems obtained by analytic continuation to complex plane of one-dimensional dynamical systems are also discussed. The evolution equations for nonlinear, amplitude-modulated Langmuir waves as well as circularly polarized electromagnetic waves in plasmas, are considered as illustrative examples.