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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.

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