J. A. Louw

Nahm’s equations, singular point analysis, and integrability

A singular point analysis (Painleve test) for certain special cases of Nahm’s equations is performed. It is shown that there are cases in which the equations do not pass the test. The Laurent expansion does not contain the right number of arbitrary expansion coefficients. Nevertheless the systems under consideration are completely integrable.

Is there a connection between local and global (in-)stability?

We review two criteria which have been used to predict the onset of large scale stochasticity in Hamiltonian systems. We show that one of them, due to Toda and based on a local stability analysis of the equations of motion, is inconclusive. An approach based on the local Riemannian curvature K of trajectories correctly predicts chaos if K < 0 everywhere, but·no further conclusions can be drawn. New (counter-)examples are provided.

Motion of Energy Levels and Energy Dependent Constants of Motion

From the eigenvalue equation ( H 0 +λ V )|ψ n (λ)>= E n (λ)|ψ n (λ)> one can derive an autonomous system of first order differential equations for the eigenvalues E n (λ) andthe matrix elements V m n (λ)= where λ is the independent variable. We investigate the case where the Hamiltonian H is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Furthermore we describe the connection with stationary state perturbation theory. Several open questions are also discussed.

Some remarks on Painlevé test and integrability

We investigate the construction of explicit space- and time-dependent soliton equations using the Painleve test. The integrability of these equations is discussed. Then we study whether or not linearizable nonlinear partial differential equations pass the Painleve test. Finally, we discuss the connection of the resonance with the conservation laws.

Singular point analysis, resonances and Yoshida's theorem

Yoshida (1983) described a connection between scale-invariant autonomous systems of ordinary differential equations, algebraic first integrals and resonances. In his analysis it is assumed that the scale invariance determines the dominant behaviour. Here the authors discuss the case where the dominant behaviour is not determined by the scale invariance. For the constructed example they also give the Lax representation.

Finite Hubbard Model with Phonon Coupling

The spectrum of the two point Hubbard model with phonon coupling is studied. In particular the connection with quantum chaos is discussed.

Quantum chaos of an exciton-phonon system

A simple model of an exciton-phonon system is studied in connection with quantum chaos.

Linearly coupled anharmonic oscillators and integrability

The Painleve test for a linearly coupled anharmonic oscillator is performed. We show that this system does not pass the Painleve test. This suggests that this system is not integrable. Moreover, we apply Ziglins (1983) theorem which provides a criterion for non-existence of first integrals besides the Hamiltonian. Calculating numerically the maximal one-dimensional Lyapunov exponent, we find regions with positive exponents. Thus, the system can show chaotic behaviour. Finally we compare our results with the quartic coupled anharmonic oscillator.