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A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation

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Abstract

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill’s equation. Applying the Evans-Krein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and track dynamical Hamiltonian–Hopf bifurcations via the Krein signature. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein–Gordon equation with a non-periodic potential.

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References

  • Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew Math. 410, 167–212 (1990)

    MathSciNet  MATH  Google Scholar 

  • Barone, A., Paternò, G.: Physics and Applications of the Josephson Effect. Wiley, Hoboken (1982)

    Book  Google Scholar 

  • Barone, A., Esposito, F., Magee, C.J., Scott, A.C.: Theory and applications of the sine-gordon equation. La Rivista del Nuovo Cimento 1(2), 227–267 (1971)

    Article  Google Scholar 

  • Bronski, J.C., Johnson, M.A., Kapitula, T.: An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type. Proc. R. Soc. Edinb. Sect. A 141(6), 1141–1173 (2011)

    Article  MathSciNet  Google Scholar 

  • Bronski, J.C., Johnson, M.A., Kapitula, T.: An instability index theory for quadratic pencils and applications. Commun. Math. Phys. 327(2), 521–550 (2014)

    Article  MathSciNet  Google Scholar 

  • Cadoni, M., Franzin, E., Masella, F., Tuveri, M.: A solution-generating method in Einstein-Scalar Gravity. Acta Applicandae Mathematicae 162(1), 33–45 (2019)

    Article  MathSciNet  Google Scholar 

  • Derks, G., Doelman, A., Knight, J.K., Susanto, H.: Pinned fluxons in a Josephson junction with a finite-length inhomogeneity. Eur. J. Appl. Math. 23(2), 201–244 (2012)

    Article  MathSciNet  Google Scholar 

  • Derks, G., Doelman, A., van Gils, S.A., Visser, T.: Travelling waves in a singularly perturbed sine-Gordon equation. Phys. D: Nonlinear Phenom. 180(1–2), 40–70 (2003)

    Article  MathSciNet  Google Scholar 

  • Derks, G., Gaeta, G.: A minimal model of DNA dynamics in interaction with RNA-Polymerase. Phys. D: Nonlinear Phenom. 240(22), 1805–1817 (2011)

    Article  MathSciNet  Google Scholar 

  • Evans, J.W.: Nerve axon equations: III stability of the nerve impulse. Indiana Univ. Math. J. 22(6), 577–593 (1972)

    Article  MathSciNet  Google Scholar 

  • Gardner, R.A.: On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72(5), 415–439 (1993)

    MathSciNet  MATH  Google Scholar 

  • Gardner, R.A.: Spectral analysis of long wavelength periodic waves and applications. J. Reine Angew Math. 491, 149–181 (1997)

    Article  MathSciNet  Google Scholar 

  • Hǎrǎguş, M., Kapitula, T.: On the spectra of periodic waves for infinite-dimensional Hamiltonian systems. Phys. D: Nonlinear Phenom. 237(20), 2649–2671 (2008)

    Article  MathSciNet  Google Scholar 

  • Jones, C.K.R.T.: Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Am. Math. Soc. 286(2), 431–469 (1984)

    Article  MathSciNet  Google Scholar 

  • Jones, C.K.R.T., Marangell, R., Miller, P.D., Plaza, R.G.: On the stability analysis of periodic sine-Gordon traveling waves. Phys. D: Nonlinear Phenom. 251, 63–74 (2013)

    Article  MathSciNet  Google Scholar 

  • Jones, C.K.R.T., Marangell, R., Miller, P.D., Plaza, R.G.: Spectral and modulational stability of periodic wavetrains for the nonlinear Klein–Gordon equation. J. Differ. Equ. 257(12), 4632–4703 (2014)

    Article  MathSciNet  Google Scholar 

  • Kapitula, T.: The Krein signature, Krein eigenvalues, and the Krein oscillation theorem. Indiana Univ. Math. J. 59(4), 1245–1275 (2010)

    Article  MathSciNet  Google Scholar 

  • Kapitula, T., Kevrekidis, P.G., Yan, D.: The Krein matrix: general theory and concrete applications in atomic Bose–Einstein condensates. SIAM J. Appl. Math. 73(4), 1368–1395 (2013)

    Article  MathSciNet  Google Scholar 

  • Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)

    MATH  Google Scholar 

  • Knobel, R.: An Introduction to the Mathematical Theory of Waves. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  • Kollár, R., Deconinck, B., Trichtchenko, O.: Direct characterization of spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems via dispersion relation. SIAM J. Math. Anal. 51(4), 3145–3169 (2019)

    Article  MathSciNet  Google Scholar 

  • Kollár, R., Miller, P.D.: Graphical Krein signature theory and Evans–Krein functions. SIAM Rev. 56(1), 73–123 (2014)

    Article  MathSciNet  Google Scholar 

  • Lafortune, S., Lega, J., Madrid, S.: Instability of local deformations of an elastic rod: numerical evaluation of the Evans function. SIAM J. Appl. Math. 71(5), 1653–1672 (2011)

    Article  MathSciNet  Google Scholar 

  • Magnus, W., Winkler, S.: Hill’s Equation. Courier Corporation, North Chelmsford (2013)

    MATH  Google Scholar 

  • Marangell, R., Miller, P.D.: Dynamical Hamiltonian–Hopf instabilities of periodic traveling waves in Klein–Gordon equations. Phys. D: Nonlinear Phenom. 308, 87–93 (2015)

    Article  MathSciNet  Google Scholar 

  • Markus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  • Natali, F.: On periodic waves for sine- and sinh-Gordon equations. J. Math. Anal. Appl. 379(1), 334–350 (2011)

    Article  MathSciNet  Google Scholar 

  • Palacios, J.M.: Orbital stability and instability of periodic wave solutions for the \(\phi ^4\)-model, Preprint (2020). arXiv:2005.09523

  • Pava, J.A., Natali, F.: (Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations. Adv. Nonlinear Anal. 3(2), 95–123 (2014)

    MathSciNet  MATH  Google Scholar 

  • Pava, J.A., Plaza, R.G.: Transverse orbital stability of periodic traveling waves for nonlinear Klein–Gordon equations. Stud. Appl. Math. 137(4), 473–501 (2016)

    Article  MathSciNet  Google Scholar 

  • Scott, A.C.: Waveform stability on a nonlinear Klein–Gordon equation. Proc. IEEE 57(7), 1338–1339 (1969)

    Article  Google Scholar 

  • Stanislavova, M., Stefanov, A.: Linear stability analysis for travelling waves of second order in time PDE’s. Nonlinearity 25(9), 2625–2654 (2012)

    Article  MathSciNet  Google Scholar 

  • Trichtchenko, O., Deconinck, B., Kollár, R.: Stability of periodic traveling wave solutions to the Kawahara equation. SIAM J. Appl. Dyn. Syst. 17(4), 2761–2783 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Peter Miller as well as the referees for their helpful suggestions for improving our paper, Dave Smith for his comments which clarified our notation, and the referees for their helpful suggestions for improving our paper. W. Clarke would like to thank Henrik Schumacher for his insightful discussion about speeding up our code. R. Marangell acknowledges the support of the Australian Research Council under Grant DP200102130.

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  1. School of Mathematics and Statistics, University of Sydney, Sydney, Australia

    W. A. Clarke & R. Marangell

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  1. W. A. Clarke
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  2. R. Marangell
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Correspondence to R. Marangell.

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Communicated by Peter Miller.

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Clarke, W.A., Marangell, R. A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation. J Nonlinear Sci 30, 3421–3442 (2020). https://doi.org/10.1007/s00332-020-09655-4

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