Communications in Mathematical Physics

, Volume 83, Issue 1, pp 83–106 | Cite as

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

  • M. Adler
  • P. van Moerbeke


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 allij, 2(N NT) ij (N N T ) jj −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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • M. Adler
    • 1
  • P. van Moerbeke
    • 2
    • 3
  1. 1.Brandeis UniversityWalthamUSA
  2. 2.University of LouvainLouvain-la-NeuveBelgium
  3. 3.Brandeis UniversityWalthamUSA

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