IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v191y2025ics0960077924013742.html
   My bibliography  Save this article

Closed-form solutions of the nonlinear Schrödinger equation with arbitrary dispersion and potential

Author

Listed:
  • Polyanin, Andrei D.
  • Kudryashov, Nikolay A.

Abstract

For the first time, the general nonlinear Schrödinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class of related nonlinear partial differential equations that are often used in various areas of theoretical physics, including nonlinear optics, superconductivity and plasma physics. To construct exact solutions, a combination of the method of functional constraints and methods of generalized separation of variables is used. New exact closed-form solutions of the general nonlinear Schrödinger equation, which are expressed in quadratures or elementary functions, are found. One-dimensional non-symmetry reductions are described, which lead the considered nonlinear partial differential equation to a simpler ordinary differential equation or a system of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for integrating nonlinear equations of mathematical physics.

Suggested Citation

  • Polyanin, Andrei D. & Kudryashov, Nikolay A., 2025. "Closed-form solutions of the nonlinear Schrödinger equation with arbitrary dispersion and potential," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:chsofr:v:191:y:2025:i:c:s0960077924013742
    DOI: 10.1016/j.chaos.2024.115822
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077924013742
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2024.115822?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
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Kudryashov, Nikolay A. & Kutukov, Aleksandr A. & Biswas, Anjan & Zhou, Qin & Yıldırım, Yakup & Alshomrani, Ali Saleh, 2023. "Optical solitons for the concatenation model: Power-law nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    2. Ekici, Mehmet & Sonmezoglu, Abdullah & Biswas, Anjan, 2021. "Stationary optical solitons with Kudryashov’s laws of refractive index," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Andrei D. Polyanin, 2020. "Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations," Mathematics, MDPI, vol. 8(1), pages 1-38, January.
    4. Zhang, Jia-Hao & Qin, Huan-Qi & Si, Zhi-Zeng & Jia, Yun-Hao & Kudryashov, Nikolay A. & Wang, Yue-Yue & Dai, Chao-Qing, 2024. "Pure-quartic soliton attracted state and multi-soliton molecules in mode-locked fiber lasers," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    5. Sucu, Nuray & Ekici, Mehmet & Biswas, Anjan, 2021. "Stationary optical solitons with nonlinear chromatic dispersion and generalized temporal evolution by extended trial function approach," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    6. Jiang, Jun-Hang & Si, Zhi-Zeng & Kudryashov, Nikolay A. & Dai, Chao-Qing & Liu, Wei, 2024. "Prediction of symmetric and asymmetric solitons and model parameters for nonlinear Schrödinger equations with competing nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    7. Ekici, Mehmet, 2022. "Kinky breathers, W-shaped and multi-peak soliton interactions for Kudryashov's quintuple power-law coupled with dual form of non-local refractive index structure," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    8. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
    9. Kudryashov, Nikolay A., 2024. "Solitons of the complex modified Korteweg–de Vries hierarchy," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    10. Kumar, Vikas & Biswas, Anjan & Ekici, Mehmet & Moraru, Luminita & Alzahrani, Abdullah Khamis & Belic, Milivoj R., 2021. "Time–dependent coupled complex short pulse equation: Invariant analysis and complexitons," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    11. Wang, Haotian & Li, Xin & Zhou, Qin & Liu, Wenjun, 2023. "Dynamics and spectral analysis of optical rogue waves for a coupled nonlinear Schrödinger equation applicable to pulse propagation in isotropic media," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    12. Alexander V. Aksenov & Andrei D. Polyanin, 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions," Mathematics, MDPI, vol. 9(4), pages 1-31, February.
    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. Anjan Biswas & Jose Vega-Guzman & Yakup Yıldırım & Luminita Moraru & Catalina Iticescu & Abdulah A. Alghamdi, 2023. "Optical Solitons for the Concatenation Model with Differential Group Delay: Undetermined Coefficients," Mathematics, MDPI, vol. 11(9), pages 1-14, April.
    2. Innocent Simbanefayi & Chaudry Masood Khalique, 2020. "Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation," Mathematics, MDPI, vol. 8(6), pages 1-20, June.
    3. Fahmy, E.S., 2008. "Travelling wave solutions for some time-delayed equations through factorizations," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1209-1216.
    4. Mustafa Inc & Rubayyi T. Alqahtani & Ravi P. Agarwal, 2023. "W-Shaped Bright Soliton of the (2 + 1)-Dimension Nonlinear Electrical Transmission Line," Mathematics, MDPI, vol. 11(7), pages 1-13, April.
    5. Chaudry Masood Khalique & Karabo Plaatjie, 2021. "Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    6. Yang, Lijuan & Du, Xianyun & Yang, Qiongfen, 2016. "New variable separation solutions to the (2 + 1)-dimensional Burgers equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1271-1275.
    7. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.
    8. Zayed, E.M.E. & Alurrfi, K.A.E., 2016. "Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 111-131.
    9. Andrei D. Polyanin, 2019. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions," Mathematics, MDPI, vol. 7(5), pages 1-19, April.
    10. Hu, Xiang & Yin, Zhixiang, 2022. "A study of the pulse propagation with a generalized Kudryashov equation," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    11. Kudryashov, Nikolay A. & Ryabov, Pavel N., 2014. "Exact solutions of one pattern formation model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1090-1093.
    12. Zdravković, S. & Zeković, S. & Bugay, A.N. & Petrović, J., 2021. "Two component model of microtubules and continuum approximation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    13. Navickas, Z. & Ragulskis, M. & Telksnys, T., 2016. "Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 333-338.
    14. Qin, Chun-Yan & Zhang, Run-Fa & Li, Yao-Hong, 2024. "Various exact solutions of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli-like equation by using bilinear neural network method," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    15. Han, Tianyong & Li, Zhao & Li, Chenyu, 2023. "Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in ," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 615(C).
    16. Nikolay A. Kudryashov & Sofia F. Lavrova, 2024. "Painlevé Analysis of the Traveling Wave Reduction of the Third-Order Derivative Nonlinear Schrödinger Equation," Mathematics, MDPI, vol. 12(11), pages 1-13, May.
    17. Nikolay A. Kudryashov, 2021. "Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 9(23), pages 1-9, November.
    18. Nickel, J., 2007. "Travelling wave solutions to the Kuramoto–Sivashinsky equation," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1376-1382.
    19. Petar Popivanov & Angela Slavova, 2024. "Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form," Mathematics, MDPI, vol. 12(7), pages 1-24, March.
    20. Ekici, Mehmet, 2022. "Kinky breathers, W-shaped and multi-peak soliton interactions for Kudryashov's quintuple power-law coupled with dual form of non-local refractive index structure," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    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:eee:chsofr:v:191:y:2025:i:c:s0960077924013742. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.