Abstract
Today, nonlinear Schrödinger-type equations are the focus of explorers and scientists. Hereby, we look into a generalized nonlinear Schrödinger equation by constructing the modified generalized Darboux transformation and analysing the type-I, type-II and type-III degenerate solitons for generalized nonlinear Schrödinger equation via some semirational solutions. Type-I degenerate solitons refer to the degenerate solitons, type-II degenerate solitons mean the interactions between the solitons and the degenerate solitons, and type-III degenerate solitons denote the bound states among a series of the degenerate solitons. We hope that the mathematical research method used in this paper could provide some theoretical assistance for future research on the nonlinear Schrödinger-type equations.
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References
Wazwaz, A.M., Albalawi, W., El-Tantawy, S.A.: Optical envelope soliton solutions for coupled nonlinear Schrödinger equations applicable to high birefringence fibers. Optik 255, 168673 (2022)
Wazwaz, A.M., El-Tantawy, S.A.: Bright and dark optical solitons for (3+1)-dimensional hyperbolic nonlinear Schrödinger equation using a variety of distinct schemes. Optik 270, 170043 (2022)
Ma, L.N., Li, S., Wang, T.M., Xie, X.Y., Du, Z.: Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan–Porsezian–Daniel equations via Riemann–Hilbert approach. Phys. Scripta 98, 075222 (2023)
Guan, W.Y., Li, B.Q.: New observation on the breather for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in inhomogeneous optical fiber. Optik 181, 853–861 (2019)
Ma, Y.L.: Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers. Nonlinear Dyn. 97, 95–105 (2019)
Ankiewicz, A., Akhmediev, N.: Higher-order integrable evolution equation and its soliton solutions. Phys. Lett. A 378, 358–361 (2014)
Su, D., Yong, X.L., Tian, Y.J., Tian, J.: Breather and rogue wave solutions of an extended nonlinear Schrödinger equation with higher-order odd and even terms. Mod. Phys. Lett. B 32, 1850309 (2018)
Zhang, R.F., Bilige, S.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)
Du, Z., Zhao, X.H.: Vector localized and periodic waves for the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. Phys. Fluids 35, 075108 (2023)
Zuo, D.W., Zhang, G.F.: Exact solutions of the nonlocal Hirota equations. Appl. Math. Lett. 93, 66–71 (2019)
Wazwaz, A.M.: Integrable (3+1)-dimensional Ito equation: Variety of lump solutions and multiple-soliton solutions. Nonlinear Dyn. 109, 1929–1934 (2022)
Kumar, S., Dhiman, S.K., Baleanu, D., Osman, M.S., Wazwaz, A.M.: Lie symmetries, closed-form solutions, and various dynamical profiles of solitons for the variable coefficient (2+1)-dimensional KP equations. Symmetry 14, 597 (2022)
Kaur, L., Wazwaz, A.M.: Optical soliton solutions of variable coefficient Biswas–Milovic (BM) model comprising Kerr law and damping effect. Optik 266, 169617 (2022)
Zhang, R.F., Li, M.C., Cherraf, A., Vadyala, S.R.: The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM. Nonlinear Dyn. 111, 8637–8646 (2023)
Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)
Wazwaz, A.M.: New (3+1)-dimensional Painlevé integrable fifth-order equation with third-order temporal dispersion. Nonlinear Dyn. 106, 891–897 (2021)
Wazwaz, A.M.: Painlevé integrability and lump solutions for two extended (3+1)-and (2+1)-dimensional Kadomtsev–Petviashvili equations. Nonlinear Dyn. 111, 3623–3632 (2023)
Salah, M., Ragb, O., Wazwaz, A.M.: Efficient discrete singular convolution differential quadrature algorithm for solitary wave solutions for higher dimensions in shallow water waves. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2022.2136420
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Wang, M., Yang, YF. Degenerate solitons in a generalized nonlinear Schrödinger equation. Nonlinear Dyn 112, 3763–3769 (2024). https://doi.org/10.1007/s11071-023-09207-x
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DOI: https://doi.org/10.1007/s11071-023-09207-x