Integrability of Nonlinear Systems and Perturbation Theory

  • V. E. Zakharov
  • E. I. Schulman
Part of the Springer Series in Nonlinear Dynamics book series (SSNONLINEAR)


The theory of so-called integrable Hamiltonian wave systems arose as a result of the inverse scattering method discovery by Gardner, Green, Kruskal and Miura [1] for the Korteveg-de Vries equation. This discovery was initiated by the pioneering numerical experiment by Kruskal and Zabusky [2]. After a pragmatic phase, which was devoted to finding new soliton equations, the theory became rather complicated. One of its branches may be called the “qualitative theory of infinite-dimensional Hamiltonian systems”, to which the results reviewed in this paper belong. We consider only Hamiltonian systems possessing Hamiltonians with a quadratic part which may be transformed in normal variables to the form
$$H_0 = \sum\limits_{\alpha = 1}^N {\omega _k ^{\left( \alpha \right)} a_k ^{\left( \alpha \right)} a_k ^{*\left( \alpha \right)} dk} $$


Nonlinear System Perturbation Theory Canonical Transformation Asymptotic State Periodic Case 
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© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • V. E. Zakharov
  • E. I. Schulman

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