A. Rauh

Global stability properties of the complex Lorenz model

Abstract By means of a quadratic Lyapunov function it is shown that the complex Lorenz model is globally stable in the sense that the trajectories asymptotically settle within a finite region around the zero fixed point. Thus, similar to the original Lorenz model, the five-dimensional complex model does not have runaway solutions. For conditions appropriate to model single mode lasers an analytical expression is given for the upper bound on the magnitude of the time-dependent electric field as a function of the model parameters (pumping rate, detuning and relaxation constants). When compared with the time-dependent solutions found by direct integration, the upper bounds exceed the maximum values reached by the asymptotic solutions by factors between 3 and 6, or between 2 and 3, for conditions below and above the second threshold, respectively. In extreme cases of transient evolution, the solutions approach within 20% of the predicted upper bounds.

The Betz optimum efficiency for windmills

The derivation of the power coefficient, cp, by Betz is critically analyzed. This is done within the hydrodynamic model used by Betz. Neither the accepted value, cp = 16/27, nor the Froude theorem can be derived rigorously from the model.

Application of normal forms to the Lorenz model in the subcritical region

Abstract It is shown that, within a certain parameter interval below the Hopf bifurcation, the Lorenz equations can be decoucpled in the normal form representation. We normalize only up to monomials of finite order and combine the unnormalized part of the flow in a remainder. The formalism avoids small denominators and is well defined if the inverse of a finite polynomial map exists. Numerical results are presented for the basin of attraction in some neighborhood of the stable fixed points and for the Hopf limit cycle.

Longitudinal sum rule for the general dielectric function

The sum rule is derived without assuming any translational or point symmetry. Within the self consistent field approximation, for the case of finite translational symmetry, a more direct proof is given.

Theory of stark ladders in the optical absorption of solids

Abstract The observation of Stark ladders in the optical absorption of poor Ohmic conductors is explained within the scheme of the time dependent Houston functions. Essentially, we overcome the restriction, used exclusively until now, that the electric field is parallel to a reciprocal lattice vector.

Analytical Localization Lengths in an One-Dimensional Disordered Electron System

Abstract Analytical approximations of the Lyapunov exponent are derived for a random displacement model with equal potential barriers and random positions of the scatterers. Two asymptotic regions are considered corresponding to high and low reflectivity of the single scattering potential. The analytical results are in terms of a distribution function W for certain phases of the transfer matrices. A functional equation for W is derived and numerically solved. This serves to validate the analytical asymptotic formulas which turn out to be accurate in the high and low reflectivity regions with dimensionless wave number K < 2 and K > 6, respectively. The high wave number asymptotics allows for an analytical examination of the sufficient conditions for Anderson localization

Finite-Time Singularities of Solutions of a Class of Nonlinear Schrodinger Equations

Solutions to the initial-boundary value problemfor a class of nonlinear Schrodinger equations areconsidered. Sufficient conditions are found so that thesolutions do not exist for all times t > 0. An explicit upper bound of the t interval ofexistence of the solutions is obtained. The timeevolution of a singularity is demonstrated numericallyin the case of the one-dimensional Schrodingerequation.

Some estimates regarding a finite time stability problem of the Sun-Earth-Jupiter system

The spatial three-body problem of the Sun, Earth, Jupiter is studied over a finite time interval comparable with the age of the solar system. Some basic concepts of Nekhoroshevs theory are adopted; however, because of the finite time horizon considered, the canonical transformation scheme can be stopped with a third-order remainder. The overall effect of Jupiter is estimated from its maximal gravitational forces acting on the orbital elements of the Earth. This is done both analytically and numerically. The conservation of energy and angular momentum are rigorously taken into account. As part of a rather extensive programme which has not yet been completed, the effects of the first-order resonances and of a typical third-order rest term are estimated. Both contributions are found to be at their largest when the two osculating ellipses are coplanar. The third order force examined is zero at opposition by time reversal symmetry, but its maximal value lies close to this constellation. It can give rise to a 5% fluctuation of the semimajor axis of the Earth, not before 12 billion years, provided the eccentricity and inclination of the Earth are confined to 0≤e≤0.2 and 0≤i≤3π/4, respectively. The self-consistent check of the two latter conditions is left to a future study. The results of this paper rely on a certain adiabatic approximation.