Kowalewski's asymptotic method, Kac-Moody lie algebras and regularization

Abstract

We use an effective criterion based on the asymptotic analysis of a class of Hamiltonian equations to determine whether they are linearizable on an abelian variety, i.e., solvable by quadrature. The criterion is applied to a system with Hamiltonian $$H = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\sum\limits_{i = 1}^l {p_i^2 } + \sum\limits_{i = 1}^{l + 1} {\exp \left( {\sum\limits_{j = 1}^l {N_{ij} x_j } } \right)} ,$$ parametrized by a real matrixN=(N ij ) of full rank. It will be solvable by quadrature if and only if for alli≠j, 2(N NT) ij (N N T ) −1 is a nonpositive integer, i.e., the interactions correspond to the Toda systems for the Kac-Moody Lie algebras. The criterion is also applied to a system of Gross-Neveu.