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Geometry of the modulational instability. III: Homoclinic orbits for the periodic sine-Gordon equation. (English) Zbl 0705.58026

Summary: [See also the authors, part I: Local analysis, part II: Global analysis. Univ. Arizona, Preprint (1987).]

The homoclinic geometric structure of the integrable sine-Gordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to N-tori. Simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given. A labeling is provided which identifies and catalogues all such orbits. These orbits are related in a one-to-one manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Bäcklund transformations.


MSC:
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37C75Stability theory
81Q05Closed and approximate solutions to quantum-mechanical equations
37A30Ergodic theorems, spectral theory, Markov operators
37K35Lie-Bäcklund and other transformations