IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2014y2014i1n848153.html

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

Author

Listed:
  • Nalan Antar

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.

Suggested Citation

  • Nalan Antar, 2014. "Pseudospectral Renormalization Method for Solitons in Quasicrystal Lattice with the Cubic‐Quintic Nonlinearity," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:848153
    DOI: 10.1155/2014/848153
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2014/848153
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2014/848153?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Jason W. Fleischer & Mordechai Segev & Nikolaos K. Efremidis & Demetrios N. Christodoulides, 2003. "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature, Nature, vol. 422(6928), pages 147-150, March.
    2. Barak Freedman & Guy Bartal & Mordechai Segev & Ron Lifshitz & Demetrios N. Christodoulides & Jason W. Fleischer, 2006. "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature, Nature, vol. 440(7088), pages 1166-1169, April.
    3. Demetrios N. Christodoulides & Falk Lederer & Yaron Silberberg, 2003. "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature, Nature, vol. 424(6950), pages 817-823, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Petroski, Miroslav M. & Timotijević, Dejan V. & Jović Savić, Dragana M. & Vasiljević, Jadranka M., 2026. "Head-on excitation of tilted solitons by fragmented Bessel beams," Chaos, Solitons & Fractals, Elsevier, vol. 202(P1).
    2. Bao, Y.Y. & Li, S.R. & Liu, Y.H. & Xu, T.F., 2022. "Gap solitons and nonlinear Bloch waves in fractional quantum coupler with periodic potential," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    3. Kartashov, Yaroslav V., 2023. "Vortex solitons in large-scale waveguide arrays with adjustable discrete rotational symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    4. Li, S.R. & Bao, Y.Y. & Liu, Y.H. & Xu, T.F., 2022. "Bright solitons in fractional coupler with spatially periodical modulated nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    5. Chen, Guanwei & Ma, Shiwang, 2014. "Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 787-798.
    6. Adrián Espínola-Rocha, J. & Kevrekidis, P.G., 2009. "Thresholds for soliton creation in the Ablowitz–Ladik lattice," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(4), pages 693-706.
    7. Zhaohui Dong & Xiaoxiong Wu & Yiwen Yang & Penghong Yu & Xianfeng Chen & Luqi Yuan, 2024. "Temporal multilayer structures in discrete physical systems towards arbitrary-dimensional non-Abelian Aharonov-Bohm interferences," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    8. Yang Yang & Robert J. Chapman & Ben Haylock & Francesco Lenzini & Yogesh N. Joglekar & Mirko Lobino & Alberto Peruzzo, 2024. "Programmable high-dimensional Hamiltonian in a photonic waveguide array," Nature Communications, Nature, vol. 15(1), pages 1-7, December.
    9. Dirk Hennig & Jesús Cuevas-Maraver, 2024. "Discrete Derivative Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 13(1), pages 1-25, December.
    10. Li, Rujiang & Kong, Xiangyu & Wang, Wencai & Jia, Yongtao & Liu, Ying, 2025. "Newton conjugate gradient method for discrete nonlinear Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
    11. Zhu, Xing & Belić, Milivoj R. & Mihalache, Dumitru & Li, Jiawei & Zeng, Liangwei, 2025. "Dark solitons in purely quintic nonlinear lattices with defects," Chaos, Solitons & Fractals, Elsevier, vol. 200(P1).
    12. Pawel S. Jung & Georgios G. Pyrialakos & Fan O. Wu & Midya Parto & Mercedeh Khajavikhan & Wieslaw Krolikowski & Demetrios N. Christodoulides, 2022. "Thermal control of the topological edge flow in nonlinear photonic lattices," Nature Communications, Nature, vol. 13(1), pages 1-7, December.
    13. Panajotov, Krassimir & Tlidi, Mustapha & Song, Yufeng & Zhang, Han, 2022. "Discrete vector light bullets in coupled χ3 nonlinear cavities," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    14. El-Nabulsi, Rami Ahmad & Anukool, Waranont, 2023. "A family of nonlinear Schrodinger equations and their solitons solutions," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    15. Peng Wang & Qidong Fu & Ruihan Peng & Yaroslav V. Kartashov & Lluis Torner & Vladimir V. Konotop & Fangwei Ye, 2022. "Two-dimensional Thouless pumping of light in photonic moiré lattices," Nature Communications, Nature, vol. 13(1), pages 1-8, December.
    16. Göksel, İzzet & Antar, Nalan & Bakırtaş, İlkay, 2018. "Two-dimensional solitons in cubic-saturable media with PT-symmetric lattices," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 83-89.
    17. Wen, Yanan & Zhao, Zhonglong & Xin, Pengcheng & Wang, Yu, 2026. "Numerical evaluations and asymptotic properties for the N-periodic wave solutions of an N=1 supersymmetric MKdV equation in fermionic field," Chaos, Solitons & Fractals, Elsevier, vol. 203(C).
    18. Chengzhi Qin & Han Ye & Shulin Wang & Lange Zhao & Menglin Liu & Yinglan Li & Xinyuan Hu & Chenyu Liu & Bing Wang & Stefano Longhi & Peixiang Lu, 2024. "Observation of discrete-light temporal refraction by moving potentials with broken Galilean invariance," Nature Communications, Nature, vol. 15(1), pages 1-11, December.
    19. Feng, Bao-Feng & Chan, Youn-Sha, 2007. "Intrinsic localized modes in a three particle Fermi–Pasta–Ulam lattice with on-site harmonic potential," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(4), pages 292-301.
    20. Shi, Zeyun & Badshah, Fazal & Qin, Lu, 2023. "Two-dimensional lattice soliton and pattern formation in a cold Rydberg atomic gas with nonlocal self-defocusing Kerr nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:848153. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.