The spectrum of travelling wave solutions to the Sine-Gordon equation

Christopher K. R. T. Jones; Robert Marangell

doi:10.3934/dcdss.2012.5.925

Article Contents

2012, Volume 5, Issue 5: 925-937. Doi: 10.3934/dcdss.2012.5.925

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The spectrum of travelling wave solutions to the Sine-Gordon equation

Christopher K. R. T. Jones^1, and
Robert Marangell^2,

1.
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599
2.
Department of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006

Received: October 2010

Revised: September 2011

Published: October 2012

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Abstract

We investigate the spectrum of the linear operator coming from the sine-Gordon equation linearized about a travelling kink-wave solution. Using various geometric techniques as well as some elementary methods from ODE theory, we find that the point spectrum of such an operator is purely imaginary provided the wave speed $c$ of the travelling wave is not $\pm 1$. We then compute the essential spectrum of the same operator.

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Mathematics Subject Classification: Primary: 35P30, 74J30 Secondary: 35P05.

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