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Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

Part of: Spectral theory and eigenvalue problems Equations and inequalities involving nonlinear operators General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  24 January 2019

Tomáš Dohnal  and
Dmitry Pelinovsky
Tomáš Dohnal
Affiliation:
Fachbereich Mathematik Technical University Dortmund, Vogelpothsweg 87, Dortmund 44221, Germany (tomas.dohnal@math.tu-dortmund.de)
Dmitry Pelinovsky
Affiliation:
Department of Mathematics, McMaster University, Hamilton, OntarioL8S 4K1, Canada (dmpeli@math.mcmaster.ca) and Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin street, Nizhny Novgorod603950, Russia
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Abstract

The stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

Keywords

bifurcationnonlinear bound statesspectral intervalsnon-selfadjoint linear Schrödinger operator𝒫𝒯-symmetry

MSC classification

Primary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 81Q12: Non-selfadjoint operator theory in quantum theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 171 - 204
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Adams, R. A. and Fournier, J. J. F.. Sobolev spaces, 2nd edn (Amsterdam: Elsevier/Academic Press, 2003).Google Scholar
2Busch, K., Schneider, G., Tkeshelashvili, L. and Uecker, H.. Justification of the nonlinear Schrödinger equation in spatially periodic media. Z. Angew. Math. Phys. 57 (2006), 905–939.CrossRefGoogle Scholar
3Conca, C. and Vanninathan, M.. Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997), 1639–1659.CrossRefGoogle Scholar
4Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
5Curtis, C. W. and Ablowitz, M. J.. On the existence of real spectra in 𝒫𝒯-symmetric honeycomb optical lattices. J. Phys. A: Math. Theor. 47 (2014), 225205 (22pp).CrossRefGoogle Scholar
6Curtis, C. W. and Zhu, Y.. Dynamics in 𝒫𝒯-symmetric honeycomb lattices with nonlinearity. Stud. Appl. Math. 135 (2015), 139–170.CrossRefGoogle Scholar
7Dohnal, T. and Siegl, P.. Bifurcation of eigenvalues in nonlinear problems with antilinear symmetry. J. Math. Phys 57 (2016), 093502.CrossRefGoogle Scholar
8Dohnal, T. and Uecker, H.. Coupled-mode equations and gap solitons for the 2D Gross–Pitaevskii equation with a non-separable periodic potential. Physica D 238 (2009), 860–879.CrossRefGoogle Scholar
9Dohnal, T., Pelinovsky, D. and Schneider, G.. Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential. J. Nonlin. Sci. 19 (2009), 95–131.CrossRefGoogle Scholar
10Eastham, M. S. P..The spectral theory of periodic differential equations. Texts in Mathematics, X, p. 130 (Edinburgh-London: Scottish Academic Press, 1973).Google Scholar
11Fibich, G.. The nonlinear Schrödinger equation: singular solutions and optical collapse. Applied Mathematical Sciences (Springer International Publishing, 2015).CrossRefGoogle Scholar
12Gelfand, I. M.. Expansion in eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk. SSSR 73 (1950), 1117–1120.Google Scholar
13Guo, A., Salamo, G. J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G. A and Christodoulides, D. N.. Observation of 𝒫𝒯-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103 (2009), 093902.CrossRefGoogle ScholarPubMed
14He, Y., Zhu, X., Mihalache, D., Liu, J. and Chen, Z.. Lattice solitons in 𝒫𝒯-symmetric mixed linear-nonlinear optical lattices. Phys. Rev. A 85 (2012), 013831.CrossRefGoogle Scholar
15Ize, J.. Bifurcation theory for Fredholm operators. Memoirs of the American Mathematical Society 7 (1976), 174.CrossRefGoogle Scholar
16Kato, T.. Perturbation theory for linear operators (Berlin: Springer–Verlag, 1995).CrossRefGoogle Scholar
17Kevrekidis, P., Pelinovsky, D. and Tyugin, D.. Nonlinear stationary states in 𝒫𝒯-symmetric lattices. SIAM. J. Appl. Dyn. Syst. 12 (2013), 1210–1236.CrossRefGoogle Scholar
18Kirrmann, P., Schneider, G. and Mielke, A.. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh A. 122 (1992), 85–91.CrossRefGoogle Scholar
19Konotop, V. V., Yang, J. and Zezyulin, D. A.. Nonlinear waves in 𝒫𝒯-symmetric systems. Rev. Modern Phys. 88 (2016), 035002 (59 pp).CrossRefGoogle Scholar
20Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H. and Christodoulides, D. N.. Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures. Phys. Rev. Lett. 106 (2011), 213901.CrossRefGoogle ScholarPubMed
21Magnus, W. and Winkler, S..Hill's equation (Dover, New York, 1977).Google Scholar
22Musslimani, Z. H., Makris, K. G., El-Ganainy, R. and Christodoulides, D. N.. Optical solitons in 𝒫𝒯 periodic potentials. Phys. Rev. Lett. 100 (2008), 030402.CrossRefGoogle ScholarPubMed
23Nixon, S. and Yang, J.. Exponential asymptotics for solitons in 𝒫𝒯-symmetric periodic potentials. Stud. Appl. Math. 133 (2014), 373–397.CrossRefGoogle Scholar
24Nixon, S., Ge, L. and Yang, J.. Stability analysis for solitons in 𝒫𝒯-symmetric optical lattices. Phys. Rev. A 85 (2012), 023822.CrossRefGoogle Scholar
25Pelinovsky, D. E.. Localization in periodic potentials: from Schrödinger operators to the Gross-Pitaevskii equation. LMS Lecture Note Series, vol. 390 (Cambridge: Cambridge University Press 2011).Google Scholar
26Pelinovsky, D. and Schneider, G.. Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential. Appl. Anal. 86 (2007), 1017–1036.CrossRefGoogle Scholar
27Reed, M. and Simon, B.. Methods of modern mathematical physics. IV. analysis of operators (New York-London: Academic Press, 1978).Google Scholar