Abstract
We discuss about the theory of nonlinear localized excitations, such as soliton and compactons in the gap of the linear spectrum of the nonlinear systems. We show how the gap originates in the linear spectrum using examples of a few systems, such as nonlinear lattices, Bose–Einstein condensates in optical lattice and systems represented by coupled nonlinear evolution equations. We then analytically show the excitation of solitons and compacton-like solutions in the gap of the linear spectrum of a system of coupled Korteweg–de Vries (KdV) equations with linear and nonlinear dispersions. Finally, we discuss about the theory of Feshbach resonance management and dispersion management of the soliton solutions.
Similar content being viewed by others
References
M Remoissenet, Waves called solitons 2nd edn (Springer-Verlag, Heidelberg, 1995)
W Chen and D L Mills, Phys. Rev. Lett. 58, 160 (1987)
W Chen and D L Mills, Phys. Rev. B 35, 524 (1987)
D L Mills and S E Trullinger, Phys. Rev. B 36, 947 (1987)
C Martijn de Sterke and J E Sipe, Phys. Rev. A 38, 5149 (1988) J E Sipe and H G Winful, Opt. Lett. 13, 132 (1988)
L Kahn, N S Almeida, and D L Mills, Phys. Rev. B 37, 8072 (1988)
S V Branis, O Martin, and J L Birman, Phys. Rev. Lett. 65, 2638 (1990)
D Barday and M Remoissenent, Phys. Rev. B 41, 10387 (1990)
D Barday and M Remoissenent, Phys. Rev. B 43, 7297 (1991)
R Grimshaw and B A Malomed, Phys. Rev. Lett. 72, 949 (1994)
O Zobay, S Potting, P Meystre, and E M Wright, Phys. Rev. A 59, 643 (1999)
V V Konotop and M Salerno, Phys. Rev. A 65, 021602 (2002)
P J Louis, E A Ostrovskaya, C M Savage, and Yu S Kivshar, Phys. Rev. A 67, 013602 (2003)
N K Efremidis and D N Christodoulides, Phys. Rev. A 67, 063608 (2003)
B Eiermann, T Anker, M Albiez, M Tagliber, P Treutlein, K P Marzlin, and M K Oberthaler, Phys. Rev. Lett. 92, 230401 (2004)
P Rosenau and J M Hyman, Phys. Rev. Lett. 70, 564 (1993)
P Rosenau, Phys. Rev. Lett. 73, 1737 (1994)
F Cooper, H Shepard, and P Sodano, Phys. Rev. E 48, 4027 (1993)
B Dey and A Khare, Phys. Rev. E 58, R2741 (1998)
B Dey and A Khare, J. Phys. A. Gen. 33, 5335 (2000)
B Dey, Phys. Rev. E 57, 4733 (1998)
P T Dinda and M Remoissenet, Phys. Rev. E 60, 6218 (1999)
M Eleftheriou, B Dey, and G P Tsironis, Phys. Rev. E 62, 7540 (2000)
B Dey, M Eleftheriou, S Flach, and G P Tsironis, Phys. Rev. E 65, 017601 (2000)
P Rosenau and S Shochet, Chaos 15, 015111 (2005)
P Rosenau and S Shochet, Phys. Rev. Lett. 94, 045503 (2005)
P Kevrekidis and V V Konotop, Phys. Rev. E 65, 066614 (2002)
R Sarkar and B Dey, J. Phys. Math. Gen. 39, L99 (2006)
R Sarkar and B Dey, Phys. Rev. E 76, 016605 (2007)
B A Malomed, Soliton management in periodic systems ( Springer, New York, 2006)
M Centurion, M Porter, P G Kevrikidis, and D Psaltis, Phys. Rev. Lett. 97, 033903 (2006)
S Flach and A V Gorbach, Phys. Rep. 467, 1 (2008)
P J Y Louis, E A Ostrovskaya, C M Savaga, and Y S Kivshar, Phys. Rev. E 67, 013602 (2003)
G A Al-Akhaly and B Dey, Phys. Rev. E 84, 036607 (2011)
Y S Kivshar and N Flytzanis, Phys. Rev. A 46, 7972 (1992)
D N Christodoulides and R I Joseph, Phys. Rev. Lett. 62, 1746 (1989)
O A Chubykalo, A S Kovalev, and O V Usatenko, Phys. Rev. B 47, 3153 (1993)
A B Aceves and S Wabnitz, Phys. Lett. A 141, 37 (1989)
S Clarke, B M Malomed, and R Grimshaw, Chaos 12, 8 (2002)
I Gabitiv and S K Turitsyn, Opt. Lett. 21, 327 (1996) M J Ablowitz and G Biondini, Opt. Lett. 23, 1668 (1998)
Zhi-Yuan Sun, Yi-Tian Gao, Ying Liu, and Yu Xin, Phys. Rev. E 84, 026606 (2011)
Acknowledgements
BD would like to thank DST and BCUD-PU for financial support through research projects.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
DEY, B. Excitations and management of the nonlinear localized gap modes. Pramana - J Phys 85, 961–974 (2015). https://doi.org/10.1007/s12043-015-1110-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-015-1110-9