Soliton Solutions of the Generalized Dullin-Gottwald-Holm Equation with Parabolic Law Nonlinearity

Abstract

In this paper, soliton solutions of the generalized Dullin-Gottwald-Holm (gDGH) equation with parabolic law nonlinearity are investigated. The gDGH describes the behavior of waves in shallow water with surface tension. There are only a few studies in the literature regarding gDGH equation with parabolic law nonlinearity, and to our best knowledge, the unified Riccati equation expansion method (UREEM) has not been applied to this equation before. Many soliton solutions of the considered gDGH equation are successfully attained using the UREEM, which is a powerful technique for solving nonlinear partial differential equations. We verify that the obtained analytical solutions satisfy the gDGH equation using Mathematica. Furthermore, some plots of the acquired solitons are demonstrated with the aid of Matlab to examine the properties of the soliton solutions. The obtained results show that the considered gDGH equation admits dark, bright, singular, and periodic solutions. This study may contribute to a comprehensive investigation of the soliton solutions of the gDGH equation, which has practical applications in fields such as oceanography and nonlinear optics.

Keywords

• Soliton solutions
• shallow water
• nonlinear wave dynamics
• dark soliton.

Parabolik Doğrusal olmayan Kanunlu Genelleştirilmiş Dullin-Gottwald-Holm Denkleminin Soliton Çözümleri

Öz

Bu makalede, parabolik yasalı genelleştirilmiş Dullin-Gottwald-Holm (gDGH) denkleminin soliton çözümleri incelenmiştir. İlgili denklem yüzey gerilimli sığ sularda dalga davranışını modellemektedir. Literatürde doğrusal olmayan parabolik kanuna sahip gDGH denklemi ile ilgili sadece birkaç çalışma vardır ve literatür araştırmalarından görüldüğü üzere bu makalede kullanılacak olan birleştirilmiş Riccati denklemi genişletme (BRDG) yöntemi daha önce bu denkleme uygulanmamıştır. Bu çalışmada gDGH denkleminin birçok soliton çözümü, doğrusal olmayan kısmi diferansiyel denklemleri çözmek için güçlü bir teknik olan BRDG yöntemi kullanılarak başarılı bir şekilde elde edilmiştir. Elde edilen analitik çözümlerin gDGH denklemini sağladığı Mathematica kullanarak doğrulanmıştır. Ayrıca, soliton çözümlerinin özelliklerini ve davranışını incelemek için elde edilen solitonların bazı grafikleri Matlab yardımıyla çizdirilmiştir. Elde edilen sonuçlar, gDGH denkleminin karanlık, parlak, tekil ve periyodik gibi farklı türlerde çözümler içerdiğini göstermektedir. Bu çalışma, okyanus bilimi ve doğrusal olmayan optik gibi alanlarda uygulamaları olan gDGH denkleminin soliton çözümlerinin kapsamlı bir şekilde incelenmesine katkı sunabilir.

Anahtar Kelimeler

• Soliton çözümleri
• sığ su
• doğrusal olmayan dalga dinamiği
• karanlık soliton.

Kaynakça

1. Biswas, A., & Kara, A. (2010). 1-Soliton solution and conservation laws of the generalizedDullin–Gottwald–Holm equation. Applied Mathematics and Computation, 217(2),929–932. https://doi.org/10.1016/j.amc.2010.05.085
2. Cakicioglu, H., Ozisik, M., Secer, A., & Bayram, M. (2023). Optical soliton solutions of Schrödinger–Hirota equation with parabolic law nonlinearity via generalized Kudryashov algorithm. Optical and Quantum Electronics, 55(5). https://doi.org/10.1007/s11082-023-04634-x
3. Dullin, H. R., Gottwald, G. A., & Holm, D. D. (2001). An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion. Physical Review Letters, 87(19). https://doi.org/10.1103/physrevlett.87.194501
4. El-Wakil, S., El-Labany, S. K., Zahran, M., & Sabry, R. (2002). Modified extended tanh-function method for solving nonlinear partial differential equations. Physics Letters, 299(2–3), 179–188. https://doi.org/10.1016/s0375-9601(02)00669-2
5. Kudryashov, N. A. (2020). Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik, 206, 163550. https://doi.org/10.1016/j.ijleo.2019.163550
6. Leta, T. D., & Li, J. (2017). Various Exact Soliton Solutions and Bifurcations of a Generalized Dullin–Gottwald–Holm Equation with a Power Law Nonlinearity. International Journal of Bifurcation and Chaos. https://doi.org/10.1142/s0218127417501292
7. Osman, M. N. M., & Wazwaz, A. (2018b). An efficient algorithm to construct multi-soliton rational solutions of the (2+ 1)-dimensional KdV equation with variable coefficients. Applied Mathematics and Computation, 321, 282–289. https://doi.org/10.1016/j.amc.2017.10.042
8. Ozisik, M., Secer, A., Bayram, M., & Aydin, H. (2022). An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices. Optik, 265, 169499. https://doi.org/10.1016/j.ijleo.2022.169499

9. Ozisik, M., Secer, A., & Bayram, M. (2022b). On the examination of optical soliton pulses of Manakov system with auxiliary equation technique. Optik, 268, 169800. https://doi.org/10.1016/j.ijleo.2022.169800
10. Ozisik, M., Secer, A., & Bayram, M. (2023). On solitary wave solutions for the extended nonlinear Schrödinger equation via the modified F-expansion method. Optical and Quantum Electronics, 55(3). https://doi.org/10.1007/s11082-022-04476-z
11. Sirendaoreji. (2017). Unified Riccati equation expansion method and its application to two new classes of Benjamin–Bona–Mahony equations. Nonlinear Dynamics, 89(1), 333–344. https://doi.org/10.1007/s11071-017-3457-6
12. Yang, D., Lou, Q., & Zhang, J. (2022). Bifurcations and exact soliton solutions for generalized Dullin–Gottwald–Holm equation with cubic power law nonlinearity. European Physical Journal Plus, 137(2). https://doi.org/10.1140/epjp/s13360-022-02462-8
13. Yıldırım, Y. (2021). Optical solitons in birefringent fibers with Biswas–Arshed equation by sine–Gordon equation method. Optik, 227, 165960. https://doi.org/10.1016/j.ijleo.2020.165960
14. Yin, J. H., Ding, S., Tian, L., & Fan, X. (2013). A New Method for Generating Traveling-Wave Solutions of Coupled Nonlinear Equations. Ukrainian Mathematical Journal, 64(10), 1553–1561. https://doi.org/10.1007/s11253-013-0734-5
15. Zayed, E. M., Biswas, A., Asma, M., Ekici, M., Alzahrani, A. K., & Belic, M. R. (2020). Pure-cubic optical soliton perturbation with full nonlinearity by unified Riccati equation expansion. Optik, 223, 165445. https://doi.org/10.1016/j.ijleo.2020.165445
16. Zhang, Y., & Xia, Y. (2021). Traveling Wave Solutions of Generalized Dullin–Gottwald–Holm Equation with Parabolic Law Nonlinearity. Qualitative Theory of Dynamical Systems, 20(3). https://doi.org/10.1007/s12346-021-00503-8
17. Zhou, Y., Wang, M., & Chen, X. (2003b). Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letters, 308(1), 31–36. https://doi.org/10.1016/s0375-9601(02)01775-9