Nonlinear wave propagation through cold plasma

Electromagnetic wave propagation through cold collision free plasma is studied using the nonlinear perturbation method. It is found that the equations can be reduced to the modified Kortweg-de Vries equation

The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems

The Fermi-Pasta-Ulam $\alpha$-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N= 32 to 128 oscillators appear to be large enough to suggest statistical mechanical behavior. A key element has been a comparison of the maximum Lyapounov coefficient $\lambda_{max}$ of the FPU $\alpha$-model and that of the Toda lattice. For generic initial conditions, $\lambda_{max}(t)$ is indistinguishable for the two models up to times that increase with decreasing energy (at fixed N). Then suddenly a bifurcation appears, which can be discussed in relation to the breakup of regular, soliton-like structures. After this bifurcation, the $\lambda_{max}$ of the FPU model appears to approach a constant, while the $\lambda_{max}$ of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU $\alpha$-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density $\epsilon_c(N)\sim 1/N^2$, below which trapping occurs; here the dynamics appears to be regular, soliton-like and the approach to equilibrium - if any - takes longer than observable on any available computer. Above this threshold the system appears to behave in accordance with statistical mechanics, exhibiting an approach to equilibrium in physically reasonable times. The initial conditions chosen by Fermi, Pasta and Ulam were not generic and below threshold and would have required possibly an infinite time to reach equilibrium.Comment: 24 pages, REVTeX, 8 PostScript figures. Published versio

Offsprings of a point vortex

The distribution engendered by successive splitting of one point vortex are considered. The process of splitting a vortex in three using a reverse three-point vortex collapse course is analysed in great details and shown to be dissipative. A simple process of successive splitting is then defined and the resulting vorticity distribution and vortex populations are analysed

Description of the inelastic collision of two solitary waves for the BBM equation

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves).Comment: submitted for publication. Corrected typo in Theorem 1.

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

Lieb-Robinson Bounds for the Toda Lattice

We establish locality estimates, known as Lieb-Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb-Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems

On the supersymmetric nonlinear evolution equations

Supersymmetrization of a nonlinear evolution equation in which the bosonic equation is independent of the fermionic variable and the system is linear in fermionic field goes by the name B-supersymmetrization. This special type of supersymmetrization plays a role in superstring theory. We provide B-supersymmetric extension of a number of quasilinear and fully nonlinear evolution equations and find that the supersymmetric system follows from the usual action principle while the bosonic and fermionic equations are individually non Lagrangian in the field variable. We point out that B-supersymmetrization can also be realized using a generalized Noetherian symmetry such that the resulting set of Lagrangian symmetries coincides with symmetries of the bosonic field equations. This observation provides a basis to associate the bosonic and fermionic fields with the terms of bright and dark solitons. The interpretation sought by us has its origin in the classic work of Bateman who introduced a reverse-time system with negative friction to bring the linear dissipative systems within the framework of variational principle.Comment: 12 pages, no figure

Energy Relaxation in Nonlinear One-Dimensional Lattices

We study energy relaxation in thermalized one-dimensional nonlinear arrays of the Fermi-Pasta-Ulam type. The ends of the thermalized systems are placed in contact with a zero-temperature reservoir via damping forces. Harmonic arrays relax by sequential phonon decay into the cold reservoir, the lower frequency modes relaxing first. The relaxation pathway for purely anharmonic arrays involves the degradation of higher-energy nonlinear modes into lower energy ones. The lowest energy modes are absorbed by the cold reservoir, but a small amount of energy is persistently left behind in the array in the form of almost stationary low-frequency localized modes. Arrays with interactions that contain both a harmonic and an anharmonic contribution exhibit behavior that involves the interplay of phonon modes and breather modes. At long times relaxation is extremely slow due to the spontaneous appearance and persistence of energetic high-frequency stationary breathers. Breather behavior is further ascertained by explicitly injecting a localized excitation into the thermalized array and observing the relaxation behavior