Abstract
In this paper, we attain the analytical soliton solutions for the two-coupled incoherent nonlinear Schrödinger equation via the Hirota bilinear method, and this equation is always used to describe the pulse propagation in high birefringent fibers. We also select different relevant parameters to construct different types of solutions including multiple-soliton, two-kink-soliton and breather solutions, which is an important tool to investigate the influence of these related parameters with different function types on solitons propagations and interactions. It is worth noting that the multiple-soliton solutions obtained are bright–dark alternating, which is not seen in previous literature. At the same time, in order to better understand the physical significances of all kinds of solutions, the 3D images are drawn with different shapes and parameters. Finally, the dynamic behaviors for two-soliton solutions before and after collision are also discussed by asymptotic analysis.
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The datasets generated during the current study are not publicly available due to individual privacy but are available from the corresponding author on reasonable request.
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)
Iqbal, A., Hamid, N.N.A., Ismail, A.I.M., Abbas, M.: Galerkin approximation with quintic B-spline as basis and weight functions for solving second order coupled nonlinear Schrödinger equations. Math. Comput. Simul. 187, 1–16 (2021)
Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)
Wazwaz, A.M.: Optical bright and dark soliton solutions for coupled nonlinear Schrödinger (CNLS) equations by the variational iteration method. Optik 207, 164457 (2020)
Azzouzi, F., Triki, H., El Akrmi, A.: Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation. Chaos Solitons Fractals 39, 1304–1307 (2009)
Melchert, O., Willms, S., Demircan, A.: (Invited) Two-color soliton meta-atoms and molecules. Optik 280, 170772 (2022)
Zhang, H.Q., Meng, X.H., Tian, B.: Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation. Phys. Scr. 75, 537–542 (2007)
Malomend: Bound solitons in coupled nonlinear Schrödinger equations. Phys. Rev. A 45, 8321–8323 (1992)
Dai, C.Q., Wang, Y.Y.: Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals. Nonlinear Dyn. 102, 1733–1741 (2020)
Fang, Y., Wu, G.Z., Dai, C.Q.: Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN. Nonlinear Dyn. 105, 603–616 (2021)
Wen, L., Guo, H., Zhang, X.F.: Effects of atom numbers on the miscibility-immiscibility transition of a binary Bose–Einstein condensate. Phys. Rev. A 101, 033610 (2020)
Cao, Q.H., Dai, C.Q.: Symmetric and anti-symmetric solitons of the fractional second and third-order nonlinear Schrödinger equation. Chin. Phys. Lett. 38, 090501 (2021)
Carr, L.D., Kutz, J.N., Reinhardt, W.P.: Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate. Phys. Rev. E 63, 066604 (2001)
Dai, C.Q., Fan, Y., Zhang, N.: Re-observation on localized waves constructed by variable separation solutions of (1+1)-dimensional coupled integrable dispersionless equations via the projective Riccati equation method. Appl. Math. Lett. 96, 20–26 (2019)
Kumar, D., Singh, J., Sushila: Analysis of regularized long-wave equation associated with a new fractional operator with Mittag–Leffler type kernel. Physica A-Siat. Mech. Appl. 492, 155–167 (2018)
Li, Z., Li, P., Han, T.Y.: Dynamical behavior and the classification of single traveling wave solutions for the coupled nonlinear Schrödinger equations with variable coefficients. Adv. Math. Phys. 2021, 9955023 (2021)
Tang, L., Chen, S.P.: The classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation. Opt. Quant. Electron. 54, 105 (2022)
Dai, C.Q., Zhang, J.F.: Controlling effect of vector and scalar crossed double-Ma breathers in a partially nonlocal nonlinear medium with a linear potential. Nonlinear Dyn. 100, 1621–1628 (2020)
Abdel-Gawada, H.I., Biswas, A., Belic, M.: Optical solitons and stability analysis with coupled nonlinear schrödinger’s equations having double external potentials. Results Physics 15, 102707 (2019)
Yang, Y.Q., Suzuki, T., Wang, J.Y.: Bäcklund transformation and localized nonlinear wave solutions of the nonlocal defocusing coupled nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. 95, 105626 (2021)
Biswas, A., Milovic, D.: Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. Numer. 15, 1473–1484 (2010)
Ji, J.L., Kai, Y., Xu, Z.W., Ma, L.Y.: On a coupled nonlocal nonlinear Schrödinger system. Chaos 164, 112761 (2022)
Dong, H.H., Wei, C.M., Fang, Y.: The Darboux transformation and N-Soliton solutions of coupled cubic-quintic nonlinear Schrödinger equation on a time-space scale. Fractal Fract. 6, 12 (2022)
Yang, J., Zhu, Y., Li, J.T.: Scalar and vector crossed breather-pair and their controlling excitations of a coupled nonlinear Schrödinger equation with partially nonlocal property in an external potential. Optik 226, 165963 (2021)
Zhu, H.P., Xu, Y.J.: High-dimensional vector solitons for a variable-coefficient partially nonlocal coupled Gross–Pitaevskii equation in a harmonic potential. Appl. Math. Lett. 124, 107701 (2022)
Yang, J., Zhu, Y., Qin, W., Wang, S.H., Li, J.T.: Spatiotemporal vector vortex and diploe solitons of a nonautonomous partially nonlocal coupled Gross–Pitaevskii equation with a linear potential. Results Phys. 30, 104860 (2021)
Degasperis, A., Lombarbo, S., Sommacal, M.: Rogue wave type solutions and spectra of coupled nonlinear Schrödinger equations. Fluids 4, 57 (2019)
Wang, X.B.: Quasi-periodic wave solutions of the nonlocal coupled nonlinear Schrödinger equation. Appl. Math. Lett. 132, 108086 (2022)
Yu, W.T., Zhang, H.X., Liu, W.J.: The collision dynamics between double-hump solitons in two mode optical fibers. Results Phys. 28, 104618 (2021)
Ramakrishnan, R., Stalin, S., Lakshmanan, M.: Multihumped nondegenerate fundamental bright solitons in N-coupled nonlinear Schrödinger system. J. Phys. A-Math. Theor. 54, 14LT01 (2021)
Musammil, N.M., Subha, P.A., Nithyanandan, K.: Phase dynamics of inhomogeneous Manakov vector solitons. Phys. Rev. E 100, 012213 (2019)
Chen, J.C., Yan, Q.X.: Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type. Nonlinear Dyn. 100, 2807–2816 (2020)
Stalin, S., Ramakrishnan, R., Lakshmanan, M.: Nondegenerate bright solitons in coupled nonlinear Schrödinger systems: recent developments on optical vector solitons. Photonics 7, 258 (2021)
Ramakrishnan, R., Stalin, S., Lakshmanan, M.: Nondegenerate solitons and their collisions in Manakov systems. Phys. Rev. E 102, 042212 (2020)
Kanna, T., Vijayajayanthi, M., Lakshmanan, M.: Coherently coupled bright optical solitons and their collisions. J. Phys. A-Math. Theor. 43, 434018 (2010)
Stalin, S., Senthilvelan, M., Lakshmanan, M.: Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions. Nonlinear Dyn. 95, 343–360 (2019)
Stalin, S., Senthilvelan, M., Lakshmanan, M.: Energy-sharing collisions and the dynamics of degenerate solitons in the nonlocal Manakov system. Nonlinear Dyn. 95, 1767–1780 (2019)
Li, J.H., Chan, H.N., Chow, K.W.: Breathers and ‘black’ rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 28, 28–38 (2015)
Yang, C.Y., Zhou, Q., Belic, M.: Bright soliton interactions in a (2+1)-dimensional fourth-order variable-coefficient nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 95, 983–994 (2019)
Cui, P.: Bilinear form and exact solutions for a new extended (2+1)-dimensional Boussinesq equation. Results Phys. 22, 103919 (2021)
Hua, Y.F., Guo, B.L., Mac, W.X., Lü, X.: Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves. Appl. Math. Model. 74, 184–198 (2019)
Na, L.: Bäcklund transformation and multi-soliton solutions for the (3+1)-dimensional BKP equation with Bell polynomials and symbolic computation. Nonlinear Dyn. 82, 311–318 (2015)
Wazwaz, A.M.: Multiple soliton solutions for (2+1)-dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations. Math. Methods Appl. Sci. 82, 1580–1586 (2011)
Rizvi, S.T.R., Seadawy, A.R., Ashra, M.A.: Multiple lump and rogue wave for time fractional resonant nonlinear Schrödinger equation under parabolic law with weak nonlocal nonlinearity. Opt. Quant. Electron. 54, 212 (2022)
Sun, Y.L., Ma, W.X., Yu, J.P.: N-soliton solutions and dynamic property analysis of a generalized three-component Hirota–Satsuma coupled KdV equation. Appl. Math. Lett. 120, 107224 (2021)
Zhang, L.L., Wang, X.M.: Periodic solitons and their interactions for a general coupled nonlinear Schrödinger system. Superlattices Microstruct. 105, 198–208 (2017)
Bi, K., Hao, H.Q., Guo, R.: Soliton, breather-like and dark-soliton-breather-like solutions for the coupled long-wave-short-wave system. Nonlinear Dyn. 108, 543–554 (2022)
Han, P.F., Bao, T.T.: Bilinear auto-Bäcklund transformations and higher-order breather solutions for the (3+1)-dimensional generalized KdV-type equation. Nonlinear Dyn. 110, 1709–1721 (2022)
Wang, C.J., Dai, Z.D., Liu, C.F.: The Breather-like and rational solutions for the integrable Kadomtsev–Petviashvili-based system. Adv. Math. Phys. 2015, 861069 (2015)
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This work was supported by the Natural Science Foundation of Shanxi (No.202103021224068).
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Yang, L., Gao, B. Solitons, kink-solitons and breather solutions of the two-coupled incoherent nonlinear Schrödinger equation. Nonlinear Dyn 112, 5621–5633 (2024). https://doi.org/10.1007/s11071-024-09336-x
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DOI: https://doi.org/10.1007/s11071-024-09336-x