The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid 'cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.

We present few-femtosecond shadowgraphic snapshots taken during the non-linear evolution of the plasma wave in a laser wakefield accelerator with transverse synchronized few-cycle probe pulses. These snapshots can be directly associated with the electron density distribution within the plasma wave and give quantitative information about its size and shape. Our results show that self-injection of electrons into the first plasma wave period is induced by a lengthening of the first plasma period. Three dimensional particle in cell simulations support our observations.Laser-wakefield accelerators (LWFA) operating in the 'bubble'-regime [1] can generate quasimonoenergetic multigigaelectronvolt electron beams [2,3] with femtosecond duration [4,5] and micrometer dimensions [6,7]. These beams are produced by accelerating electrons in laser-driven plasma waves over centimeter distances. They have the potential to be compact alternatives to conventional accelerators [8]. In a LWFA, the short driving laser pulse displaces plasma electrons from the stationary background ions. The generated space charge fields cause the electrons to oscillate and form a plasma wave in the laser's wake. This wave follows the laser at almost c, the speed of light; for low amplitude it has a wavelength ofwhere n e is the electron density of the plasma. At high amplitude, electrons from the background can be injected into the wake and accelerated, producing monoenergetic electron pulses [9][10][11]. Significant progress has been made regarding achievable peak energy [3], beam stability [12] and the generation of bright X-ray pulses [13][14][15]. Until now, most of our knowledge about the dynamics of the self-injection process has been derived from detailed particle-in-cell (PIC) simulations. These simulations show that self-focusing [16] and pulse compression [17] play a vital role in increasing the laser pulse intensity prior to injection. Furthermore, simulations indicate that self-injection of electrons is associated with a dynamic lengthening of the first plasma wave's period (the 'bubble'). This lengthening can be driven by changes of the electric field structure inside the plasma wave caused by the injected electrons [18]. In contrast, the lengthening may also be due to an intensity amplification of the laser pulse caused by the non-linear evolution of the plasma wave [19,20] or due to a local increase in intensity caused by two colliding pulses [21]. In these latter scenarios, injection is a consequence of the lengthening of the bubble. However, experimental insight into these processes is extremely challenging due to the small spatial and temporal scales of a LWFA.The plasma wave, a variation in the electron density, has an associated refractive index profile which can be detected using longitudinal [22][23][24] or transverse probes [5]. Longitudinal probes cannot measure the rapid and dynamic evolution of the plasma wave that occurs in nonlinear wakefield accelerators and suffer from the strong refraction caused by the steep refractive ...

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the "first Fourier mode slice," a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example, Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of traveling waves, are uncovered.

The effective increase of the critical density associated with the interaction of relativistically intense laser pulses with overcritical plasmas, known as self-induced transparency, is revisited for the case of circular polarization. A comparison of particle-in-cell simulations to the predictions of a relativistic cold-fluid model for the transparency threshold demonstrates that kinetic effects, such as electron heating, can lead to a substantial increase of the effective critical density compared to cold-fluid theory. These results are interpreted by a study of separatrices in the single-electron phase space corresponding to dynamics in the stationary fields predicted by the cold-fluid model. It is shown that perturbations due to electron heating exceeding a certain finite threshold can force electrons to escape into the vacuum, leading to laser pulse propagation. The modification of the transparency threshold is linked to the temporal pulse profile, through its effect on electron heating.

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the 'physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors.