American Institute of Mathematical Sciences

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

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

Aslihan Demirkaya 1, , Panayotis G. Kevrekidis 2, , Milena Stanislavova 3, and  Atanas Stefanov 4,

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.
Keywords: standing waves., linear stability.
Mathematics Subject Classification: Primary: 35B35, 35B40, 35L70; Secondary: 37L10, 37L15, 37D1.
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), 947-1018.

[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), 103904; Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Optical Solitons in $\mathcal{PT}$ Periodic Potentials Phys. Rev. Lett., 100 (2008), 030402.

[3]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Unidirectional nonlinear PT-symmetric optical structures, Phys. Rev. A, 82 (2010), 043803.

[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), L171.

[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), 093902.

[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), 192.

[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), 040101.

[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), 062122.

[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), 173.

[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), 394.

[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), 234101.

[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), 023203.

[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), DOI: 10.1111/sapm.12053.

[14]

P. G. Kevrekidis, Variational method for nonconservative field theories: Formulation and two $\mathcal{PT}$-symmetric examples, Phys. Rev. A, 89 (2014), 010102(R).

[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), 224101.

[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), 1-13.

[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), 783.

[18]

M. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.

[19]

M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.

[20]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.

show all references

References:
[1]

C. M. Bender, Making Sense of Non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947-1018.

[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), 103904; Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Optical Solitons in $\mathcal{PT}$ Periodic Potentials Phys. Rev. Lett., 100 (2008), 030402.

[3]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Unidirectional nonlinear PT-symmetric optical structures, Phys. Rev. A, 82 (2010), 043803.

[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), L171.

[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), 093902.

[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), 192.

[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), 040101.

[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), 062122.

[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), 173.

[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), 394.

[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), 234101.

[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), 023203.

[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), DOI: 10.1111/sapm.12053.

[14]

P. G. Kevrekidis, Variational method for nonconservative field theories: Formulation and two $\mathcal{PT}$-symmetric examples, Phys. Rev. A, 89 (2014), 010102(R).

[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), 224101.

[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), 1-13.

[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), 783.

[18]

M. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.

[19]

M. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.

[20]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.

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