### 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) |
---|---|

Pages (from-to) | 349-384 |

Number of pages | 36 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 43 |

Issue number | 2-3 |

DOIs | |

State | Published - 1990 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

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

**Geometry of the modulational instability. III. Homoclinic orbits for the periodic sine-Gordon equation.** / Ercolani, N.; Forest, M. G.; McLaughlin, David W.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 43, no. 2-3, pp. 349-384. https://doi.org/10.1016/0167-2789(90)90142-C

}

TY - JOUR

T1 - Geometry of the modulational instability. III. Homoclinic orbits for the periodic sine-Gordon equation

AU - Ercolani, N.

AU - Forest, M. G.

AU - McLaughlin, David W.

PY - 1990

Y1 - 1990

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001522381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001522381&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(90)90142-C

DO - 10.1016/0167-2789(90)90142-C

M3 - Article

VL - 43

SP - 349

EP - 384

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 2-3

ER -