Sine-Gordon wobbles through Bäcklund transformations

Sara Cuenda; Niurka R. Quintero; Angel Sánchez

doi:10.3934/dcdss.2011.4.1047

Article Contents

2011, Volume 4, Issue 5: 1047-1056. Doi: 10.3934/dcdss.2011.4.1047

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Sine-Gordon wobbles through Bäcklund transformations

1.
Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Análisis Económico: Economía Cuantitativa, Universidad Autónoma de Madrid, Francisco Tomás y Valiente 5, 28049, Cantoblanco, Madrid
2.
Departamento de Física Aplicada I, E. U. P., Universidad de Sevilla, Virgen de África 7, 41011, Sevilla
3.
Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid

Received: August 31, 2009

Revised: September 30, 2009

Published: September 2011

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Abstract

In this work we construct the wobble exact solution of sine-Gordon equation by means of Bäcklund Transformations. We find the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular $3$-soliton solutions. We show that this solution agrees with the wobbles obtained by Kälbermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that define the building blocks of this solution -- a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation.

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Mathematics Subject Classification: 37K35, 37K40, 70K75.

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