Abstract
We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier–Bloch decomposition and the implicit function theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a nondegeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross–Pitaevskii equation approximated by solutions of the coupled-mode equations are obtained for a finite-time interval.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aceves, A.B., Costantini, B., De Angelis, C.: Two-dimensional gap solitons in a nonlinear periodic slab waveguide. J. Opt. Soc. Am. B 12, 1475–1479 (1995)
Aceves, A.B., Fibich, G., Ilan, B.: Gap-soliton bullets in waveguide gratings. Physica D 189, 277–286 (2004)
Agueev, D., Pelinovsky, D.: Modeling of wave resonances in low-contrast photonic crystals. SIAM J. Appl. Math. 65, 1101–1129 (2005)
Aközbek, N., John, S.: Optical solitary waves in two and three dimensional photonic bandgap structures. Phys. Rev. E 57, 2287–2319 (1998)
Arnold, V.: Remarks on perturbation theory for problems of Mathieu type. Usp. Mat. Nauk 38, 189–203 (1983) (English translation: Russ. Math. Surv. 38, 215–233 (1983))
Brazhnyi, V.A., Konotop, V.V., Kuzmiak, V., Shchesnovich, V.S.: Nonlinear tunneling in two-dimensional lattices. Phys. Rev. A 76, 023608 (2007)
Busch, K., Schneider, G., Tkeshelashvili, L., Uecker, H.: Justification of the nonlinear Schrödinger equation in spatially periodic media. Z. Angew. Math. Phys. 57, 905–939 (2006)
Dohnal, T., Aceves, A.B.: Optical soliton bullets in (2+1)D nonlinear Bragg resonant periodic geometries. Stud. Appl. Math. 115, 209–232 (2005)
de Sterke, C.M., Sipe, J.E.: Gap solitons. Prog. Opt. 33, 203 (1994)
de Sterke, C.M., Salinas, D.G., Sipe, J.E.: Coupled-mode theory for light propagation through deep nonlinear gratings. Phys. Rev. E 54, 1969–1989 (1996)
Eastham, M.S.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh (1973)
Gelfand, I.M.: Expansion in eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk. SSSR 73, 1117–1120 (1950)
Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, Berlin (1985)
Goodman, R.H., Weinstein, M.I., Holmes, P.J.: Nonlinear propagation of light in one-dimensional periodic structures. J. Nonlinear Sci. 11, 123–168 (2001)
Harrison, J.M., Kuchment, P., Sobolev, A., Winn, B.: On occurrence of spectral edges for periodic operators inside the Brillouin zone. J. Phys. A: Math. Theor. 40, 7597–7618 (2007)
Heinz, H.P., Küpper, T., Stuart, C.A.: Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation. J. Differ. Equ. 100, 341–354 (1992)
Kuchment, P.: The mathematics of photonic crystals. In: Mathematical Modeling in Optical Science. SIAM, Philadelphia (2001)
Küpper, T., Stuart, C.A.: Necessary and sufficient conditions for gap-bifurcation. Nonlinear Anal. 18, 893–903 (1992)
Mills, D.L.: Nonlinear Optics: Basic Concepts. Springer, Berlin (1984)
Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
Parnovski, L.: Bethe–Sommerfeld conjecture. Ann. Henri Poincaré 9, 457–508 (2008)
Pelinovsky, D., Schneider, G.: Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential. Appl. Anal. 86, 1017–1036 (2007)
Pelinovsky, D.E., Sukhorukov, A.A., Kivshar, Yu.S.: Bifurcations and stability of gap solitons in periodic potentials. Phys. Rev. E 70, 036618 (2004)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)
Schneider, G., Uecker, H.: Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems. Asymptot. Anal. 28, 163–180 (2001)
Shi, Z., Yang, J.: Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media. Phys. Rev. E 75, 056602 (2007)
Stuart, C.A.: Bifurcations into spectral gaps. Bull. Belg. Math. Soc. Simon Stevin (1995), suppl., 59pp
Weinstein, M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–67 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.I. Weinstein.
Rights and permissions
About this article
Cite this article
Dohnal, T., Pelinovsky, D. & Schneider, G. Coupled-Mode Equations and Gap Solitons in a Two-Dimensional Nonlinear Elliptic Problem with a Separable Periodic Potential. J Nonlinear Sci 19, 95–131 (2009). https://doi.org/10.1007/s00332-008-9027-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-008-9027-9
Keywords
- Gross–Pitaevskii equation
- Coupled-mode equations
- Existence of gap solitons
- Fourier–Bloch transform
- Lyapunov–Schmidt reductions