Toward a Topological Classification of Integrable PDE’s

  • Nicholas M. Ercolani
  • David W. McLaughlin
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 22)


We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both of its arguments provides a transparent realization of a Bott function and of the remaining building blocks of the stratification; in this manner, Fomenko’s structure theorems are expressed through the inverse spectral transform. Thus, soliton equations are shown to provide natural representatives of the classification in the context of PDE’s.


Double Point Integrable Hamiltonian System Soliton Equation Topological Classification Critical Circle 
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  1. 1.
    A.T. Fomenko and H. Zieschang, Topological Classification of Integrable Hamilto- nian Systems, Preprint France (1988), IHES; Criterion of Topological Equivalence of Integrable Hamiltonian Systems with Two Degrees of Freedom, Izvestiya Akad. Nauk SSSR (1990).Google Scholar
  2. 2.
    A.T. Fomenko, and H. Zieschang, Symplectic Topology of Completely Integrable Hamiltonian Systems, Uspechi Matem. Nauk 44, No. 1 (1989), 145–173.MATHGoogle Scholar
  3. 3.
    N. Ercolani, M.G. Forest, and D.W. McLaughlin, Geometry of the Modulational Instability, Memoirs of the AMS (to appear); Physica D 43 (1990), 349–384.MathSciNetMATHGoogle Scholar
  4. 4.
    M.G. Forest and D.W. McLaughlin, Modulations of Sinh-Gordon and Sine-Gordon Wavetrains, Studies in Appl. Math. 68 (1983), 11–59.MathSciNetMATHGoogle Scholar
  5. 5.
    N. Ercolani and M.G. Forest, The Geometry of Real Sine-Gordon Wavetrains, Commun. Math. Phys. 99 (1985), 1–49.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    E. Overman and C. Schober. Private communications (1989).Google Scholar
  7. 7.
    M.S. Ablowitz and B.M. Herbst, On Homoclinic Boundaries in the Nonlinear Schroedinger Equation, Proc. CRM Workshop on Hamiltonian Systems, ed. by J. Hamad and J.E. Marsden, CRM Publication, Univ. Montreal (1989).Google Scholar
  8. 8.
    E. Previato, Hyperelliptic Quasi-periodic and Soliton Solutions of the Nonlinear-Schroedinger Equation, Duke Math. J. 52 (1985), p. 329.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A. Bishop, M.G. Forest, D.W. McLaughlin, and E.A. Overman, Model Representations of Chaotic Attractors for the Driven Damped Pendulum Chain, Phys. Lett. A144 (1990), 17–25.Google Scholar
  10. 10.
    A.R. Bishop, D.W. McLaughlin, M.G. Forest, and E.A. Overman II, Quasi-periodic Route to Chaos in a Near Integrable PDE: Homoclinic Crossings, Phys. Lett. A127 (1988), 335–340.MathSciNetGoogle Scholar
  11. 11.
    H.T. Moon, Homoclinic Crossings and Pattern Selection, Phys. Rev. Lett. 64 (1990), 412–414.CrossRefGoogle Scholar
  12. 12.
    G. Kovacic and S. Wiggins, Orbits Homoclinic to Resonances: Chaos in a Model of the Forced and Damped Sine-Gordon Equation, Preprint, Cal. Inst. Tech. (1989).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Nicholas M. Ercolani
    • 1
  • David W. McLaughlin
    • 2
  1. 1.Department of MathematicsUnivesity of ArizonaTucsonUSA
  2. 2.Department of Mathematics Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

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