E. W. Laedke

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.

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.

ON LOCALIZED SOLUTIONS IN NONLINEAR FARADAY RESONANCE

The dynamics of a nonlinear modulated cross-wave of resonant frequency ω1 and carrier frequency ω ≈ ω 1 is considered. The wave is excited in a long channel of width 6 that contains water of depth d, which is subjected to a vertical oscillation of frequency 2ω. As has been shown by Miles (1984 b ), the complex amplitude satisfies a cubic Schrodinger equation with weak damping and parametric driving. The stability of its solitary wave solution is considered here in various parameter regions. We find that in a certain regime the solitary wave is stable. Completely new is the result of instability outside this parameter regime. The instability has also been verified numerically. It is shown that the final stage of solitary wave instability is a cnoidal-wave-type solution.

Stability of activation-barrier-lowering solitons

Abstract A simple one-dimensional, nonlinear, scalar kink-bearing system is investigated. The possible types of kink solutions as well as the stability of the latter are discussed. The results have applications in biological and solid state systems.

Liapunov stability of generalized Langmuir solitons

The stability investigations for small amplitude Langmuir solitons are generalized to finite amplitude solitary waves. Previous stability considerations for Langmuir envelope solitons are based on the cubic nonlinear Schrodinger equation or the so‐called Zakharov equations. Thus, they are only valid in the weakly nonlinear regime. To discuss the longitudinal stability of finite amplitude solitary waves a more general low‐frequency response has to be allowed for. Taking the full nonlinear ion equations, the stability behavior of finite amplitude solitary waves is investigated. The method is completely nonlinear and makes use of Liapunov theory. A stability criterion is derived which proves the longitudinal stability for the known stationary wave solutions. The role of transverse instabilities is also discussed.

Nonlinear ion-acoustic waves in weak magnetic fields

The two‐dimensional dynamics of nonlinear ion‐acoustic waves in plasmas are considered. A new nonlinear equation is found which is valid for unmagnetized as well as magnetized plasmas. For exactly vanishing magnetic fields, the Kadomtsev–Petviashvili equation is recovered; for weak magnetic fields, however, the dynamics are qualitatively different from that equation, depending on amplitude. With increasing magnetic field, the new equation becomes similar (but not identical) to the Zakharov–Kuznetsov equation which was derived for very strong magnetic fields. The stability behavior of plane solitons propagating along the magnetic field is also discussed. The transition from stable to unstable behavior and some quantitative new results for the instability growth rates are presented.

Dynamical properties of drift vortices

Drift vortices are described by the Hasegawa–Mima equation in the case of density gradient drift waves. From computer results, it is known that the modon solutions of the Hasegawa–Mima equation are quite stable. Here the linear stability of ground state modon solutions is proved analytically by an indirect technique.

Two-dimensional dynamics of relativistic solitons in cold plasmas

The two-dimensional dynamics of solitons appearing during relativistic laser-plasma interaction is investigated. The analysis starts from known soliton models in one space-dimension (1D). Some of the soliton solutions are already unstable in 1D, and all suffer from transverse instability in two dimensions (2D). The most unstable modes are calculated. They give a hint to the 2D structures which appear because of transversal effects. The linear stability considerations are supplemented by full 2D nonlinear simulations.

Optimization of dispersion-managed optical fiber lines

We present various approaches to the optimization of optical fiber lines and discuss the ranges of validity of such methods. An effective scheme for upgrading of existing transmission lines using dispersion-management with optimization of the pre- and post-compensating fiber is examined. The theory and numerical methods are illustrated in application to the upgrade of a specific installed Deutsche Telekom fiber line.

Drift vortices in inhomogeneous plasmas: Stationary states and stability criteria

This paper contains a theoretical treatment of the dynamical behavior of two‐dimensional nonlinear drift waves in plasmas. A simple model is investigated that allows gradients in temperature as well as in particle number density. The general class of stationary states can be specified and from the general description simplifed states can be (re)derived. A quite general method, resulting in general criteria, is proposed to study the dynamics of drift vortices under perturbations. The results are applied to several cases.