K. H. Spatschek

Evolution theorem for a class of perturbed envelope soliton solutions

Envelope soliton solutions of a class of generalized nonlinear Schrodinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.

Brillouin backscattering instability in magnetized plasmas

The decay of a circularly polarized electromagnetic wave propagating parallel to an external magnetic field into another circularly polarized electromagnetic wave and an ion acoustic wave in a homogeneous plasma is considered.

Path-averaged chirped optical soliton in dispersion-managed fiber communication lines

We present a comprehensive path-average theory of dispersion-managed (DM) optical pulse. Applying complete basis of the chirped Gauss-Hermite orthogonal functions, we derive a path-average propagation equation in the time domain and present an analytical description of the breathing dynamics of the chirped DM soliton. This theory describes both self-similar evolution of the central, energy-containing core and accompanying nonstationary oscillations of the far-field tails of an optical pulse propagating in a fiber line with an arbitrary dispersion map. In the case of a strong dispersion management the DM soliton is well described by a few modes in this expansion, justifying the use of a Gaussian trial function in the previously developed variational approach. Suggested expansion in the basis of chirped Gauss-Hermite functions presents a regular way to describe soliton properties for arbitrary dispersion map and to account for the effect of practical perturbations (filters, gratings, noise an so on) on the dynamics of the ideal DM soliton. We also present path-averaged propagation model in the spectral domain that could be useful for multichannel transmission applications. Theoretical results are verified by numerical simulations.

Variational approach to optical pulse propagation in dispersion compensated transmission systems

Within the area of optical pulse propagation in long-haul transmission systems various designs for dispersion compensation are investigated. On the basis of variational procedures with collective coordinates, a very effective method is presented which allows to determine quite accurately the possible operation points. We have obtained an analytical formula for the soliton power enhancement. This analytical expression is in good agreement with numerical results, in the (practical) limit when residual dispersion and nonlinearity only slightly affect the pulse dynamics over one compensation period. The procedure is suitable to analyze the proper design of dispersion compensating elements. The results allow also to describe the shape of the dispersion-managed soliton. We discuss also a qualitative physical explanation of the possibility to transmit a soliton at zero or normal average dispersion. Analytical predictions are confirmed by direct numerical simulations.

High resolution numerical studies of separatrix splitting due to non-axisymmetric perturbation in DIII-D

In DIII-D the splitting and deformation of the separatrix due to externally applied resonant magnetic perturbations is calculated using a vacuum field line integration code (TRIP3D–MAFOT). The resulting footprint pattern on the divertor target plates is shown in high resolution by contour plots of the connection lengths and penetration depths of the magnetic field lines. Substructures inside the divertor footprint stripes are discovered. Regions of deep penetrating long connecting field lines, which are related to the internal resonances by their manifolds, alternate with regions of regular short connecting field lines. The latter are identified as compact laminar flux tubes, which perforate the perturbed plasma region close to the x-point. The properties and consequences of such flux tubes are investigated in detail. The interaction of different resonant magnetic perturbations is analysed considering the separatrix manifolds. Constructive and destructive interference of the manifolds is discovered and studied.

Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system

Asymptotical and mapping methods to study the structure of magnetic field perturbations and magnetic field line dynamics in a tokamak ergodic divertor in toroidal geometry are developed. The investigation is applied to the Dynamic Ergodic Divertor under construction for the Torus Experiment for the Technology Oriented Research (TEXTOR-94) Tokamak at Julich [Fusion Eng. Design 37, 337 (1997)]. An ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge is considered. In cylindrical geometry, the analytical expressions for the vacuum magnetic field perturbations of such a coil system are derived, and its properties are studied. Corrections to the magnetic field due to the toroidicity are presented. The asymptotical analysis of transformation of magnetic perturbation into the Hamiltonian perturbation in toroidal geometry is carried out, and the asymptotic formulas for the spectrum of the Hamiltonian perturbations are found. A new method of integration of Hamiltonian equa...

Two-dimensional drift vortices and their stability

Drift vortices in plasmas described by the Petviashvili equation in the case of strong temperature inhomogeneities or by the Hasegawa–Mima equation in the case of density gradients are investigated. Both equations allow for two‐dimensional vortex solutions. The models are reviewed and the forms of the vortices are discussed. In the temperature‐gradient case, the stationary solutions are only known numerically, whereas in the density gradient case analytical expressions exist. The latter are called modons; here the ground states are investigated. The result of a stability calculation is that both types of two‐dimensional solutions, for the Petviashvili equation as well as the Hasegawa–Mima equation, are stable. The methods used to prove this result are either direct (constructing Liapunov functionals) or indirect, and then based on variational principles.

Twist mapping for the dynamics of magnetic field lines in a tokamak ergodic divertor

Symplectic twist mapping is proposed to model magnetic field line dynamics in the ergodic divertor at the tokamak plasma edge. The relationship between a perturbation function in the mapping and magnetic field perturbation in the tokamak is found. The mapping is specified for the Dynamic Ergodic Divertor being proposed for the Torus Experiment for Technology Oriented Research (TEXTOR-94) [Fusion Eng. Design, 37, 337 (1997)]. The spectrum of the poloidal harmonics of perturbation is assumed to be localized around the harmonics m=12. It creates the stochastic layer near the resonant magnetic surface q=3. The mapping is applied to the formation of the stochastic layer and field line diffusivity at the plasma edge. For the moderate magnetic field perturbations, the ergodic layer consists of a stochastic sea with regular Kolmogorov–Arnold–Mozer (KAM) -stability islands. The radial profiles of the Kolmogorov lengths and the field line diffusivity are studied for different perturbations. It is shown that the beh...

Footprint structures due to resonant magnetic perturbations in DIII-D

Numerical modeling of the typical footprint structures on the target plates of a divertor tokamak is presented. In the tokamak DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] toroidal mode number n=3 resonant magnetic perturbations are responsible for characteristic footprint stripes. The numerics can resolve substructures within each footprint stripe, which are related to the internal magnetic topology. It is shown that the footprint structures on the inner target plate can be predicted by the unstable manifolds of the separatrix and the q=4 resonant surface. By their intersection with the divertor target plate the unstable manifolds form the footprint boundary and substructures within. Based on the manifold analysis, the boundaries and interior structures of the footprints are explained. A direct connection of all magnetic resonances inside the stochastic plasma volume to the target plates is verified.

Traces of stable and unstable manifolds in heat flux patterns

Experimental observations of heat fluxes on divertor plates of tokamaks show typical structures (boomerang wings) for varying edge safety factors. The heat flux patterns follow from general principles of nonlinear dynamics. The pattern selection is due to the unstable and stable manifolds of the hyperbolic fixed points of the last intact island chain. Based on the manifold analysis, the experimental observations can be explained in full detail. Quantitative results are presented in terms of the penetration depths of field lines.