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The method of Poisson pairs in the theory of nonlinear PDEs

  • Franco Magri
  • Gregorio Falqui
  • Marco Pedroni
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 632)

Abstract

The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg–de Vries (KdV) equation.

In these lectures we touch the following subjects:

i) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;

ii) the theoretical basis of the method of Poisson pairs;

iii) the Gel'fand–Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;

iv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite–dimensional Grassmannian manifold.

v) the reduction technique(s) and its use to construct classes of solutions;

vi) the role of the technique of separation of variables in the study of the reduced systems;

vii) some relations intertwining the method of Poisson pairs with the method of Lax pairs.

Keywords

Poisson Bracket Darboux Transformation Hamiltonian Vector Poisson Manifold mKdV Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  • Franco Magri
    • 1
  • Gregorio Falqui
    • 2
  • Marco Pedroni
    • 3
  1. 1.Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Via degli Arcimboldi 8, I-20126 MilanoItaly
  2. 2.SISSA, Via Beirut 2/4, I-34014 TriesteItaly
  3. 3.Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 GenovaItaly

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