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

N. Ercolani; M. G. Forest; David W. McLaughlin

doi:10.1016/0167-2789(90)90142-C

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

N. Ercolani
, M. G. Forest
, David W. McLaughlin

Mathematics

Research output: Contribution to journal › Article › peer-review

123 Scopus citations

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	https://doi.org/10.1016/0167-2789(90)90142-C
State	Published - Jul 1990

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(90)90142-C

Cite this

@article{b0527a4b7ee744968e1a04d13b399262,

title = "Geometry of the modulational instability. III. Homoclinic orbits for the periodic sine-Gordon equation",

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 B{\"a}cklund transformations.",

author = "N. Ercolani and Forest, \{M. G.\} and McLaughlin, \{David W.\}",

note = "Funding Information: We wish to thank Professor Hermann Flaschka, in general, for many extremely useful discussions and in particular for his suggestion to use Biicklund theory for the construction of solutions associated to double points in the spectral transform. We also thank Professor Edward Overman for many useful numerical studies without which we could not have developed the theory. Research support from NSF, AFOSR and ONR, under contract numbers DMS-8703397, DMS 8411002, AFOSR-F4962086C0130, N00014-85-K-0412, is gratefully acknowledged. In addition, one of us (N. Ercolani) acknowledges support from the Sloan Foundation as well as an NSF postdoctoral fellowship.",

year = "1990",

month = jul,

doi = "10.1016/0167-2789(90)90142-C",

language = "English (US)",

volume = "43",

pages = "349--384",

journal = "Physica D: Nonlinear Phenomena",

issn = "0167-2789",

publisher = "Elsevier B.V.",

number = "2-3",

}

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.

N1 - Funding Information: We wish to thank Professor Hermann Flaschka, in general, for many extremely useful discussions and in particular for his suggestion to use Biicklund theory for the construction of solutions associated to double points in the spectral transform. We also thank Professor Edward Overman for many useful numerical studies without which we could not have developed the theory. Research support from NSF, AFOSR and ONR, under contract numbers DMS-8703397, DMS 8411002, AFOSR-F4962086C0130, N00014-85-K-0412, is gratefully acknowledged. In addition, one of us (N. Ercolani) acknowledges support from the Sloan Foundation as well as an NSF postdoctoral fellowship.

PY - 1990/7

Y1 - 1990/7

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 - https://www.scopus.com/pages/publications/0001522381

UR - https://www.scopus.com/pages/publications/0001522381#tab=citedBy

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

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

M3 - Article

AN - SCOPUS:0001522381

SN - 0167-2789

VL - 43

SP - 349

EP - 384

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 2-3

ER -

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

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