# American Institute of Mathematical Sciences

October  2012, 5(5): 925-937. doi: 10.3934/dcdss.2012.5.925

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

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

Received  October 2010 Revised  September 2011 Published  January 2012

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.
Citation: Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
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