# American Institute of Mathematical Sciences

2015, 2015(special): 359-368. doi: 10.3934/proc.2015.0359

## Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation

 1 Department of Mathematics, University of Hartford, 200 Bloomfield Avenue, West Hartford, CT 06117, United States 2 Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, United States 3 Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045–7523 4 Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received  September 2014 Revised  April 2015 Published  November 2015

In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.
Citation: Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
##### References:
 [1] C. M. Bender, Making Sense of Non-Hermitian Hamiltonians,, Rep. Prog. Phys., 70 (2007), 947. Google Scholar [2] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam Dynamics in $\mathcal{PT}$ Symmetric Optical Lattices,, Phys. Rev. Lett., 100 (2008). Google Scholar [3] H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Unidirectional nonlinear PT-symmetric optical structures,, Phys. Rev. A, 82 (2010). Google Scholar [4] A. Ruschhaupt, F. Delgado, and J. G. Muga, Physical realization of a $\mathcal{PT}$-symmetric potential scattering in a planar slab waveguide,, J. Phys. A: Math. Gen., 38 (2005). Google Scholar [5] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of $\mathcal{PT}$-symmetry breaking in complex optical potentials,, Phys. Rev. Lett., 103 (2009). Google Scholar [6] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of paritytime symmetry in optics,, Nature Phys., 6 (2010). Google Scholar [7] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries,, Phys. Rev. A, 84 (2011). Google Scholar [8] H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, Bypassing the bandwidth theorem with $\mathcal{PT}$ symmetry,, Phys. Rev. A, 85 (2012). Google Scholar [9] C. M. Bender, B. J. Berntson, D. Parker and E. Samuel, Observation of $\mathcal{PT}$-phase transition in a simple mechanical system,, Am. J. Phys., 81 (2013). Google Scholar [10] B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Paritytime-symmetric whispering-gallery microcavities,, Nature Physics, 10 (2014). Google Scholar [11] N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Observation of Asymmetric Transport in Structures with Active Nonlinearities,, Phys. Rev. Lett., 110 (2013). Google Scholar [12] A. Demirkaya, D. J. Frantzeskakis, P. G. Kevrekidis, A. Saxena, and A. Stefanov, Effects of $\mathcal{PT}$-symmetry in Nonlinear Klein-Gordon Field Theories and Their Solitary Waves,, Phys. Rev. E, 88 (2013). Google Scholar [13] A. Demirkaya, M. Stanislavova, A. Stefanov, T. Kapitula, and P. G. Kevrekidis, On the Spectral Stability of Kinks in Some $\mathcal{PT}$-Symmetric Variants of the Classical KleinGordon Field Theories,, Studies Appl. Math., (2014). Google Scholar [14] P. G. Kevrekidis, Variational method for nonconservative field theories: Formulation and two $\mathcal{PT}$-symmetric examples,, Phys. Rev. A, 89 (2014). Google Scholar [15] J. Cuevas, L. Q. English, P.G. Kevrekidis, and M. Anderson, Discrete Breathers in a Forced-Damped Array of Coupled Pendula: Modeling, Computation, and Experiment,, Phys. Rev. Lett., 102 (2009). Google Scholar [16] M. Stanislavova and A. Stefanov, Spectral stability analysis for special solutions of second order in time PDE's: the higher dimensional case,, Physica D, 262 (2013), 1. Google Scholar [17] M. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation,, Radiophys. Quantum Electron., 16 (1973). Google Scholar [18] M. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. Google Scholar [19] M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. Google Scholar [20] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations,, Trans. Amer. Math. Soc., 290 (1985), 701. Google Scholar

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##### References:
 [1] C. M. Bender, Making Sense of Non-Hermitian Hamiltonians,, Rep. Prog. Phys., 70 (2007), 947. Google Scholar [2] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam Dynamics in $\mathcal{PT}$ Symmetric Optical Lattices,, Phys. Rev. Lett., 100 (2008). Google Scholar [3] H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Unidirectional nonlinear PT-symmetric optical structures,, Phys. Rev. A, 82 (2010). Google Scholar [4] A. Ruschhaupt, F. Delgado, and J. G. Muga, Physical realization of a $\mathcal{PT}$-symmetric potential scattering in a planar slab waveguide,, J. Phys. A: Math. Gen., 38 (2005). Google Scholar [5] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of $\mathcal{PT}$-symmetry breaking in complex optical potentials,, Phys. Rev. Lett., 103 (2009). Google Scholar [6] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of paritytime symmetry in optics,, Nature Phys., 6 (2010). Google Scholar [7] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries,, Phys. Rev. A, 84 (2011). Google Scholar [8] H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, Bypassing the bandwidth theorem with $\mathcal{PT}$ symmetry,, Phys. Rev. A, 85 (2012). Google Scholar [9] C. M. Bender, B. J. Berntson, D. Parker and E. Samuel, Observation of $\mathcal{PT}$-phase transition in a simple mechanical system,, Am. J. Phys., 81 (2013). Google Scholar [10] B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Paritytime-symmetric whispering-gallery microcavities,, Nature Physics, 10 (2014). Google Scholar [11] N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Observation of Asymmetric Transport in Structures with Active Nonlinearities,, Phys. Rev. Lett., 110 (2013). Google Scholar [12] A. Demirkaya, D. J. Frantzeskakis, P. G. Kevrekidis, A. Saxena, and A. Stefanov, Effects of $\mathcal{PT}$-symmetry in Nonlinear Klein-Gordon Field Theories and Their Solitary Waves,, Phys. Rev. E, 88 (2013). Google Scholar [13] A. Demirkaya, M. Stanislavova, A. Stefanov, T. Kapitula, and P. G. Kevrekidis, On the Spectral Stability of Kinks in Some $\mathcal{PT}$-Symmetric Variants of the Classical KleinGordon Field Theories,, Studies Appl. Math., (2014). Google Scholar [14] P. G. Kevrekidis, Variational method for nonconservative field theories: Formulation and two $\mathcal{PT}$-symmetric examples,, Phys. Rev. A, 89 (2014). Google Scholar [15] J. Cuevas, L. Q. English, P.G. Kevrekidis, and M. Anderson, Discrete Breathers in a Forced-Damped Array of Coupled Pendula: Modeling, Computation, and Experiment,, Phys. Rev. Lett., 102 (2009). Google Scholar [16] M. Stanislavova and A. Stefanov, Spectral stability analysis for special solutions of second order in time PDE's: the higher dimensional case,, Physica D, 262 (2013), 1. Google Scholar [17] M. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation,, Radiophys. Quantum Electron., 16 (1973). Google Scholar [18] M. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. Google Scholar [19] M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. Google Scholar [20] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations,, Trans. Amer. Math. Soc., 290 (1985), 701. Google Scholar
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