Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

International Journal of Phytoremediation

Volumn 86, Issue 8, 2007, Pages 1017-1036

  Pelinovsky, Dmitry ^a   Schneider, Guido ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

2000 Mathematics Subject Classifications; 35C20; 35L60; 35Q55; 41A60; Coupled mode equations; Fourier analysis; Gap solitons in periodic potentials; Justification of asymptotic approximations; Lyapunov Schmidt reductions

Indexed keywords

EID: 52449105521     PISSN: 15226514     EISSN: 15497879     Source Type: Journal    
DOI: 10.1080/00036810701493850     Document Type: Article

References (22)

1. Agueev, D, and Pelinovsky, D. 2005. Modeling of wave resonances in low-contrast photonic crystals. SIAM Journal of Applied Mathematics, 65: 1101–1129.
2. Alama, S, and Li, YY. 1992. Existence of solutions for semilinear elliptic equations with indefinite linear part. Journal of Differential Equations, 96: 89–115.
3. Alfimov, G, and Konotop, VV. 2000. On the existence of gap solitons. Physica D, 146: 307–327.
4. Busch, K, Schneider, G, Tkeshelashvili, L, and Uecker, H. 2006. Justification of the nonlinear Schrödinger equation in spatially periodic media. Zeitschrift fiir Angew andte Mathematik und Physik, 57: 905–939.
5. Chen, H. 2004. Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations. Nonlinearity, 17: 2041–2056.
6. Chugunova, M, and Pelinovsky, D. 2006. Block-diagonalization of the symmetric first-order coupled-mode system. SIAM Journal of Applied Dynamical Systems, 5: 66–83.
7. Craig, W, and Nicholls, DP. 2000. Traveling two and three dimensional capillary gravity water waves. SIAM Journal of Mathematical Analysis, 32: 323–359.
8. Craig, W, and Wayne, CE. 1993. Newton's method and periodic solutions of nonlinear wave equations. Communications on Pure and Applied Mathematics, 46: 1409–1498.
9. Eleonsky, VM, Kulagin, NE, Novozhilova, NS, and Silin, VP. 1988. Solutions of wave equations that are self-localized in space and time-periodic. Selecta Mathematica Sovietica, 7: 1–14.
10. Heinz, HP, Kupper, T, and Stuart, CA. 1992. Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation. Journal of Differential Equations, 100: 341–354.
11. Georgieva, A, Kriecherbauer, T, and Venakides, S. 1999. Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium. SIAM Journal of Applied Mathematics, 60: 272–294.
12. Goodman, RH, Weinstein, MI, and Holmes, PJ. 2001. Nonlinear propagation of light in one-dimensional periodic structures. Journal of Nonlinear Science, 11: 123–168.
13. Groves, MD, and Schneider, G. 2001. Modulating pulse solutions for a class of nonlinear wave equations. Communications in Mathematical Physics, 219: 489–522.
14. Groves, MD, and Schneider, G. 2005. Modulating pulse solutions for quasilinear wave equations. Journal of Differential Equations, 219: 221–258.
15. Kato, T. 1976. Perturbation Theory for Linear Operators, New York: Springer-Verlag.
16. Kevrekidis, PG, Franzeskakis, DJ, and Carretero–Gonzalez, R. 2007. Emergent Nonlinear Phenomena in Bose–Einstein Condensates, New York: Springer–Verlag.
17. Kuchment, P. 1999. “ The mathematics of photonic crystals ”. In Mathematical Modeling in Optical Sciences, Philadelphia: SIAM.
18. Kupper, T, and Stuart, CA. 1992. Necessary and sufficient conditions for gap-bifurcation. Nonlinear Analysis, 18: 893–903.
19. Pankov, A. 2005. Periodic nonlinear Schrdinger equation with application to photonic crystals. Milan Journal of Mathematics, 73: 259–287.
20. Pelinovsky, DE, Sukhorukov, AA, and Kivshar, YS. 2004. Bifurcations and stability of gap solitons in periodic potentials. Physical Review E, 70: 036618 1–17.
21. Schneider, G, and Uecker, H. 2001. Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems. Asymptotic Analysis, 28: 163–180.
22. de Sterke, CM, and Sipe, JE. 1994. Gap solitons. Progress in Optics, 33: 203–259.