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

**by
Victor Z. Enolskii**

Institute of Magnetism,

Kiev, Ukraine

and Friday Februar 12, 1999, 14.00 h

at MIDIT, Building 305, room 027, DTU

**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.

Tuesday February 9, 1999, 15.00 h

at OFD, Risø

The Meeting Room

in ELM, Bldg. 109

**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.