Graduate School in Nonlinear Science

Sponsored by the Danish Research Academy

MIDIT                               OFD                           CATS
Modelling, Nonlinear Dynamics       Optics and Fluid Dynamics     Chaos and Turbulence Studies
and Irreversible Thermodynamics     Risø National Laboratory      Niels Bohr Institute and 
Technical University of Denmark     Building 128                  Department of Chemistry
Building 321                        P.O. Box 49                   University of Copenhagen 
DK-2800 Lyngby                      DK-4000 Roskilde              DK-2100 Copenhagen Ø
Denmark                             Denmark                       Denmark

Abstract: Algebraic curve appear naturally as spectral variety, when we work with the Lax representation of the completely integrable dynamical systems. But only in the case, when the curve is elliptic and in the simplest cases of hyperelliptic curve it is easy to paramtrise the dynamical variable (integrate the system) in terms of the functions which are associated with the curve - abelian functions.

There exist a number realizations of abelian functions; we concentrate on the realization, which was originated by Weierrstrass and Klein and represents itself the natural generalization of the Weierstrass theory of elliptic functions to arbitrary curve. The Kleinian theory of abelian functions is very well fit to study completely integrable equations of the KdV type, which arise as differential relations between the basis functions.

In the first lecture we remind the procedure of inversion of elliptic and hyperelliptic integrals and consider in detals the first nontrivial example of Kleinian functions of algebraic curve, i.e. ultraelliptic Kleinian functions. We shortly discuss the remarcable Kummer quatic surface, which is an element of the Kleinian theory of abelian functions. We also show, how arises integrable hierachies - KdV and ``sine-Gordon" hierarchies as well as stationary flow of the Veselov-Novikov equation.

In the second lecture we develop the theory for arbitrary curve and aimed to solve the Jacobi inversion problem in terms of Kleinian functions. The principal working tool for this is the Weierstrass gap theorem and the fundamental object of the construction is the Kleinian sigma-function, which generalizes to the case of arbitrary algebraic curve the principal properties of the sigma-function of the Weierstrass theory of elliptic functions. The Kleinian sigma-function is generating for Kleinian zeta and p-functions for which we derive the principal relations. These relations generailze the addition theorems of the Weierestrass theory of elliptic functions and yield in particular, the solution of the Jacobi inversion problem, what is the cornerstone in finding trajectories of dynamical systems.

To demonstrate the application of the developed theory we execute the integration in terms of hyperelliptic Kleinian function of the Lattice KdV equation.

Abstract: We consider nonintegrable coupled Schrödinger equations and introduce special ansatz, which narrows the class of solution, but the system is reduced to the integrable dynamical system of two particles interacting with quartic potential. The system is integrated in terms of Kleinian hyperelliptic functions, which represent themselve a natural generalisation of the Weierestrass elliptic functions. Further we consider the reduction of these hyperelliptic function to elliptic ones, which is yielded by some constrains on the parameters of the problem. As the result we derive a special solution in terms of elliptic function to the problem of wave propagation in the connected wave-guids.