Darboux coordinates and Liouville-Arnold integration in loop algebras

Abstract

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part g + 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 Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. The results are formulated in terms of sheaves to allow for singularities due to a degenerate spectrum. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits, reduced by the group of constant loops. A similar construction involving a framing at infinity is given for the nonreduced orbits. The general construction is given for g = gl(r) or sl(r), with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. As illustrative examples, the case g = 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, as well as the quasi-periodic solutions of the cubically nonlinear Schrödinger equation. For g = sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation. This case requires a further symplectic constraining procedure in order to deal with singularities in the spectral data at ∞.

Effective Inverse Spectral Problem for Rational Lax Matrices and Applications

“…These formulae provide the integration of any isospectral dynamics on rational matrix-valued functions available in the literature on integrable systems (see, e.g. [14,1,2,28] and references therein) and are, in fact, even more general. Indeed if η depends (smoothly) on one or several a "time" parameters, the above formulae would provide integration of the flow on the isospectral manifold induced by the dependence of η.…”

“…Note that the framework we are proposing is more general than the one in [1,2] since the differential η need not have poles coinciding with any of the poles of X (which is the case in loc. cit.…”

“…The forward problem and beautiful connections with integrable systems were explored in [3,14,27,28,1,2,25,21,22,23,24,16] (to name a few); in particular it was shown in these works the important rôle of the theory of Theta functions [15] in the solution of both forward and inverse problems.…”

“…Orthogonal polynomials p n (x) with respect to a weight on the real line, w N (x) = e −N V (x) dx, satisfy a three-term recurrence relation 5 [30] xp n (x) = γ n p n+1 (x) + β n p n (x) + γ n−1 p n−1 (x) , (1)(2) which can be written in matrix form as xp = Qp, with p the semiinfinite vector of the orthogonal polynomials and Q the tri-diagonal (symmetric) matrix with entries given by the coefficients of the above recurrence relations.…”

Algebraic Complete Integrability of the Bloch–Iserles System

“…We shall use the following definitions (compare with [1,2,6,8] The system (1.1) is equivalent to the following Lax pair system with parameter (see [4]…”