### Abstract

In this paper 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.

Original language | English (US) |
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Pages (from-to) | 349-384 |

Number of pages | 36 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 43 |

Issue number | 2-3 |

DOIs | |

State | Published - Jul 1990 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

Ercolani, N., Forest, M. G., & McLaughlin, D. W. (1990). Geometry of the modulational instability. III. Homoclinic orbits for the periodic sine-Gordon equation.

*Physica D: Nonlinear Phenomena*,*43*(2-3), 349-384. https://doi.org/10.1016/0167-2789(90)90142-C