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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 848153, 17 pages
http://dx.doi.org/10.1155/2014/848153
Research Article

Pseudospectral Renormalization Method for Solitons in Quasicrystal Lattice with the Cubic-Quintic Nonlinearity

Department of Mathematics, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey

Received 31 March 2014; Revised 21 July 2014; Accepted 22 July 2014; Published 10 September 2014

Academic Editor: Keshlan S. Govinder

Copyright © 2014 Nalan Antar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We modified the spectral renormalization method as the pseudospectral renormalization method in order to find the localized solutions. The pseudospectral renormalization method can be applied to a large class of problems including different homogeneities. Using this computational method, we demonstrate the existence of two different solitons in optical media described by the self-focusing cubic and the self-defocusing quintic nonlinear Schrödinger equation with quasicrystal lattice. It is shown that there are two different lattice solitons corresponding to the first and the second renormalization factors for the self-focusing cubic and the self-defocusing quintic model. However, the self-focusing quintic nonlinearity without optical lattice does not support two different solitons. We showed that the lattice solitons corresponding to the first and the second renormalization factors have the same powers and amplitudes. We also demonstrate that quintic nonlinearity supports bistable solitons by adding the optical lattice such as a quasicrystal lattice. The linear and nonlinear stabilities of these solitons are investigated using direct simulation of the nonlinear Schrödinger equation with the cubic-quintic nonlinearity and its linearized equation.