Dynamics and geometry of integrable systems with deficiency
by
Simonetta Abenda
Università di Bologna, Italy
Coauthors: Yuri Fedorov (Universita Politecnica de Catalunya, Barcelona)

Finite-dimensional reductions of the shallow water (Camassa-Holm) equation and the Dym-type equation and generalizations of the integrable Henon-Heiles and the Neumann systems are significative examples of integrable systems with deficiency. Such notion of integrability generalizes that of algebraic complete integrability. Indeed an integrable system with deficiency is a real Hamiltonian system integrable in the usual Arnold-Lioville sense, whose n-dimensional Liouville tori are extended to n-dimensional nonlinear subvarieties, W_n, of Abelian tori of dimension g >= n. Nonlinearity means that such subvarieties are not complex Abelian groups. In addition, we require that on universal coverings of the Abelian tori there exist coordinate j₁...j_g such that under the Hamiltonian flow j₁, ..., j_n change linearly in time whereas the rest of the coordinates are analytic functions of the former.

In this talk, I present an overview of the algebro-geometrical and dynamical characterization of such integrable systems obtained in collaboration with Yu. Fedorov with emphasis to the case in which W_n is an n-dimensional stratum of a (generalized) hyperelliptic Jacobian.

First of all, we give an algebraic geometrical interpretation of the weak Kowalevski-Painlevé property. Indeed we obtain exact estimates for the number and leading behaviour of principal balances of n-point symmetric functions. Principal and secondary balances, for these functions, correspond to intersections and tangency of the flow on W_n with lower dimensional strata along which the local projection of W_n to Cⁿ = (j₁, ..., j_n) ramifies.

Moreover, we consider analogs of the Jacobi-Mumford systems (coordinates and vector fields) on the natural fibering over the base formed by genus g hyperelliptic curves \Gamma whose fibers are n-dimensional strata W_n of Jac (\Gamma) (n <= g). At any point D of W_n, n independent invariant vector fields can be constructed which span the tangent space of W_n at D. The vector fields we construct on W_n are restrictions of the usual Jacobi-Mumford ones on Jac(\Gamma). It turns out that integrable systems with deficiency may be viewed as constrained algebraically completely integrable Hamiltonians according to the Dirac formalism.

Finally, we extend the notion of integrability with deficiency to discrete systems. Indeed the classical ellipsoidal billiard is an integrable map and its restriction to generic invariant manifolds (hyperelliptic Jacobians) are represented by shifts, which can be interpreted in terms of addition formulae for hyperelliptic functions.

On the other hand, there exists an infinite hierarchy of integrable ellipsoidal billiards with polynomial potentials separable in ellipsoidal coordinates. We show that, in contrast to the classical potentianless billiard and the billiard with the Hooke potential, the generic invariant manifolds of the separable potential billiards are non-abelian subvarieties of hyperelliptic Jacobians. We present an algebraic geometrical description of such maps and its representation in a discrete Lax form. Finally, in the case of quartic potentials, we establish a connection with billiards on quadrics.

References

[1] S. Abenda, `The Mumford representation for a hyperlliptically separable system with deficiency', p. 1-9, in `Proceedings of the International Conference SPT2002', eds. S. Abenda, G. Gaeta and S. Walcher, World Scientific (2002),

[2] S. Abenda and Yu. Fedorov, `On the weak Kowalewski-Painlevé property for hyperelliptically separable systems', Acta Appl. Math. v. 60, p. 137-178 (2000);

[3] S. Abenda and Yu. Fedorov, 'Ellipsoidal billiards with separable polynomial potentials and billiards on quadrics', in press (2003),

[4] M. Adler and P. van Moerbeke, `The complex geometry of the Kowalevski-Painlevé analysis', Invent. Math. v. 97, p. 3-51 (1989),

[5] M.S. Alber, R. Camassa, Yu. Fedorov, D.D. Holm, J.E. Marsden, `The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type' Comm. Math. Phys. v. 221 p. 197-227, (2001),

[6] Yu. Fedorov, `The billiard with the quadratic potential' Funct. Anal. Appl. v. 35, p. 55-66 (2001),

[7] J. Moser, A.P. Veselov, `Discrete versions of some classical integrable system and factorization of matrix polynomials', Commun. Math. Phys. v. 139, p. 217-243 (1991),

[8] P. Vanhaecke, `Integrable systems and symmetric products of curves', Math. Z. v. 227, p. 93-127 (1998).

Date received: May 26, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-92.