# 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 and Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
##### References:
 [1] A. Abbondandolo, "Morse Theory for Hamiltonian Systems," Chapman & Hall/CRC Research Notes in Mathematics, 425, Chapman & Hall/CRC, Boca Raton, FL, 2001. [2] V. I. Arnol'd, On a characteristic class entering into conditions of quantization, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861. [3] P. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in "Dynamics Reported," Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989. [4] J. C. Bronski and M. A. Johnson, Krein signatures for the Faddeev-Takhtajan eigenvalue problem, Communications in Mathematical Physics, 288 (2009), 821-846. doi: 10.1007/s00220-009-0777-5. [5] R. Buckingham and P. Miller, Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation, Physica D, 237 (2008), 2296-2341. doi: 10.1016/j.physd.2008.02.010. [6] F. Magee, C. J. Barone, A. Esposito and A. Scott, Theory and applications of the sine-Gordon equation, Riv. Nuovo. Cimento, 1 (1971), 227-267. [7] G. Derks, A. Doelman, S. A. van Gils and T. Visser, Travelling waves in a singularly perturbed sine-Gordon equation, Physica D, 180 (2003), 40-70. doi: 10.1016/S0167-2789(03)00050-2. [8] V. Maslov, "Theory of Perturbations and Asymptotic Methods," French translation of Russian original, 1965, 1972. [9] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [10] J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W. [11] M. Salerno, Discrete model for DNA-promoter dynamics, Physical Review A (3), 44 (1991), 5292-5297. doi: 10.1103/PhysRevA.44.5292. [12] A. Scott, F. Chu and D. McLaughlin, The soliton: A new concept in applied science, Proc. of the IEEE, 61 (1973), 1443-1483. doi: 10.1109/PROC.1973.9296. [13] A. Scott, Waveform stability on a nonlinear Klein-Gordon equation, Proc. Letters of the IEEE, (1969). [14] A. Scott, F. Chu and S. Reible, Magnetic-flux propagation on a Josephson transmission line, J. Applied Phys., 47 (1976), 3272-3286. doi: 10.1063/1.323126. [15] G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.

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##### References:
 [1] A. Abbondandolo, "Morse Theory for Hamiltonian Systems," Chapman & Hall/CRC Research Notes in Mathematics, 425, Chapman & Hall/CRC, Boca Raton, FL, 2001. [2] V. I. Arnol'd, On a characteristic class entering into conditions of quantization, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861. [3] P. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in "Dynamics Reported," Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989. [4] J. C. Bronski and M. A. Johnson, Krein signatures for the Faddeev-Takhtajan eigenvalue problem, Communications in Mathematical Physics, 288 (2009), 821-846. doi: 10.1007/s00220-009-0777-5. [5] R. Buckingham and P. Miller, Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation, Physica D, 237 (2008), 2296-2341. doi: 10.1016/j.physd.2008.02.010. [6] F. Magee, C. J. Barone, A. Esposito and A. Scott, Theory and applications of the sine-Gordon equation, Riv. Nuovo. Cimento, 1 (1971), 227-267. [7] G. Derks, A. Doelman, S. A. van Gils and T. Visser, Travelling waves in a singularly perturbed sine-Gordon equation, Physica D, 180 (2003), 40-70. doi: 10.1016/S0167-2789(03)00050-2. [8] V. Maslov, "Theory of Perturbations and Asymptotic Methods," French translation of Russian original, 1965, 1972. [9] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [10] J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W. [11] M. Salerno, Discrete model for DNA-promoter dynamics, Physical Review A (3), 44 (1991), 5292-5297. doi: 10.1103/PhysRevA.44.5292. [12] A. Scott, F. Chu and D. McLaughlin, The soliton: A new concept in applied science, Proc. of the IEEE, 61 (1973), 1443-1483. doi: 10.1109/PROC.1973.9296. [13] A. Scott, Waveform stability on a nonlinear Klein-Gordon equation, Proc. Letters of the IEEE, (1969). [14] A. Scott, F. Chu and S. Reible, Magnetic-flux propagation on a Josephson transmission line, J. Applied Phys., 47 (1976), 3272-3286. doi: 10.1063/1.323126. [15] G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.
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