M. R. Adams

Darboux coordinates and Liouville-Arnold integration in loop algebras

AbstractDarboux coordinates are constructed on rational coadjoint orbits of the positive frequency part

Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loop algebras

\tilde{\mathfrak{g}}^+

Dual moment maps into loop algebras

of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.

The Krichever map, vector bundles over algebraic curves, and Heisenberg algebras

AbstractThe approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad*-invariant Poisson submanifolds of the dual

The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications

(\widetilde{gl}(r)^ + )*

The evolution of fidelity in sensory systems

of the positive part of the loop algebra