We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov- Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov- Witten classes and their descendents. SISSA Preprint 65/2001/FM

this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro–geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1 − λP0) and (P2 − µP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The frameworks is applied to Lax equations with spectral parameter, for which not only it unifies the separation techniques of Sklyanin and of Magri, but also provides a more efficient “inverse ” procedure not involving the extraction of roots. 1

This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in the present paper may be found in Bloch, Crouch, Marsden and Ratiu [1998].

this paper, as well as in the works by Ol'shanetsky and Perelomov [OP], Reyman and Semenov-Tian-Shansky [RSTS1], Symes [Sy], the method of the explicit integration of the Toda equations was extended to the case when evolution takes place on the dual space of the Borel subalgebra of g . This space is foliated into symplectic leaves of different dimensions and the natural question is what can be said about the Liouville complete integrability of the Toda flows on each of these leaves. In the particular case of generic symplectic leaves in sl(n) the complete integrability was proved by Deift, Li, Nanda and Tomei [DLNT]. This paper was motivated by the work [DLNT] and its Lie algebraic interpretation proposed in [S1], [S2], [EFS]. Our main result is the following

Abstract. We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac–Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from [FFR] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P 1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.

OF THE DISSERTATION Lagrangian and symplectic techniques in discrete mechanics by James William Gilliam Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, August, 1996 Professor John C. Baez, Chairperson By a "discrete mechanical system," we mean a system in which time evolution proceeds in integer steps and the configuration space is a finite set. The most widely studied examples are cellular automata on finite lattices. We develop algebraic analogs of many of the basic concepts from differential geometry to overcome obstacles created by working over algebraic structures. We extend the methods of Lagrangian mechanics to treat discrete mechanical systems. In particular, we derive an algebraic analog of the Euler-Lagrange equation for discrete mechanics, starting from a variational principle. As an example of how this analog works, we prove a version of Noether's theorem applicable to this context. We relate our framework to Hamiltonian mechanics,...

Abstract. We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac–Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra bg should be encoded by affine opers associated to the Langlands dual affine algebra L bg. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schrödinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality. 1.

The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigid-body or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasi-phase-matching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.