Optical dual-waves to a new dual-mode extension of a third order dispersive nonlinear Schrödinger’s equation. (English) Zbl 1527.81044
Summary: The dual-mode equations refer to nonlinear models that elucidate motion of bi-directional waves traveling simultaneously under the influence of enclosed phase velocity. The initial two-mode model was introduced by S. V. Korsunsky [Phys. Lett., A 185, No. 2, 174–176 (1994; Zbl 0959.35504)], who refined the Korteweg-De Vries equation (KDVe) into a second-order form. In this study, we aim to extend nonlinear Schrödinger equation by restructuring it into a dual-mode format, and subsequently examine the geometric assessment of this new model. The extended exponential function expansion scheme, tanh-coth method, and Kudryashov method are utilized to obtain bi-directional explicit solutions. Additionally, we extensively examine impact of phase velocity on the propagation behavior of these paired waves, utilizing 2D and 3D graphs for analysis. The solutions obtained in this study have significant implications for the propagation of solitons in nonlinear optics. As examined model appears in various applications, all the derived solutions can contribute to the interpretation of the underlying mechanisms behind many nonlinear phenomena in different fields, such as, nonlinear optics, plasma physics, Bose-Einstein condensates, and others.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 30B50 | Dirichlet series, exponential series and other series in one complex variable |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 82D10 | Statistical mechanics of plasmas |
| 81V73 | Bosonic systems in quantum theory |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
Keywords:
nonlinear Schrödinger’s equation; dual-mode waves; tanh-coth method; Kudryashov method; extended exponential-expansion schemeCitations:
Zbl 0959.35504References:
| [1] | Kadomtsev, B. B.; Petviashvili, V. I., On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 1, 539-541 (1970) · Zbl 0217.25004 |
| [2] | Davey, A.; Stewartson, K., On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A, 338, 101-110 (1974) · Zbl 0282.76008 |
| [3] | Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423 |
| [4] | Akinyemi, L.; Nisar, K. S.; Saleel, C. A.; Rezazadeh, H.; Veeresh, P.; Khater, M. M.A.; Inc, M., Novel approach to the analysis of fifth-order weakly nonlocal fractional Schrödinger equation with Caputo derivative, Results Phys., 31, Article 104958 pp. (2021) |
| [5] | Zafar, A.; Shakeel, M.; Ali, A.; Akinyemi, L.; Rezazadeh, H., Optical solitons of nonlinear complex Ginzburg-Landau equation via two modified expansion schemes, Opt. Quantum Electron., 54, 5 (2022) |
| [6] | Houwe, A.; Abbagari, S.; Inc, M.; Gambo, B., Envelope solitons of the nonlinear discrete vertical dust grain oscillation in dusty plasma crystals, Chaos Solitons Fractals, 155, Article 111640 pp. (2022) |
| [7] | Abbagari, S.; Saliou, Y.; Houwe, A.; Akinyemi, L.; Inc, M.; Bouetou, T. B., Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearity, Phys. Lett. A, 442, Article 128191 pp. (2022) · Zbl 1496.81047 |
| [8] | Halidou, H.; Abbagari, S.; Houwe, A.; Inc, M.; Bouetou, T. B., Rational W-shape solitons on a nonlinear electrical transmission line with Josephson junction, Phys. Lett. A, 430, Article 127951 pp. (2022) · Zbl 1486.81096 |
| [9] | Rahman, R. U.; Raza, N.; Jhangeer, A.; Inc, M., Analysis of analytical solutions of fractional Date-Jimbo-Kashiwara-Miwa equation, Phys. Lett. A, 470, 1, Article 128773 pp. (2023) · Zbl 1522.35146 |
| [10] | Ahmad, S.; Saifullah, S.; Khan, Arshad; Inc, M., New local and nonlocal soliton solutions of a nonlocal reverse space-time mKdV equation using improved Hirota bilinear method, Phys. Lett. A, 450, Article 128393 pp. (2022) · Zbl 1511.35078 |
| [11] | Ali, K. K.; Raheel, M.; Inc, M., Some new types of optical solitons to the time-fractional new Hamiltonian amplitude equation via extended Sinh-Gorden equation expansion method, Phys. Lett. B, 36, Article 2250089 pp. (2022) |
| [12] | Ping, Y. Z.; Ping, Z. W., Self-rapping of three-dimensional spatiotemporal solitary waves in self-focusing Kerr media, Chin. Phys. Lett., 29, Article 064211 pp. (2012) |
| [13] | Zhong, W. P.; Belić, M. R.; Huang, T., Two-dimensional accessible solitons in PT-symmetric potentials, Nonlinear Dyn., 70, 2027-2034 (2012) |
| [14] | Zhong, W. P.; Belić, M. R.; Malomed, B. A.; Zhang, Y.; Huang, T., Spatiotemporal accessible solitons in fractional dimensions, Phys. Rev. E, 94, Article 012216 pp. (2016) |
| [15] | Yang, Z.; Zhong, W. P.; Belić, M.; Zhang, Y., Controllable optical rogue waves via nonlinearity management, Opt. Express, 26, 6, 7587-7597 (2018) |
| [16] | Jaradat, I.; Alquran, M.; Momani, S.; Biswas, A., Dark and singular optical solutions with dual-mode nonlinear Schrödinger’s equation and Kerr-law nonlinearity, Optik, 172, 822-825 (2018) |
| [17] | Korsunsky, S. V., Soliton solutions for a second-order KdV equation, Phys. Lett. A, 185, 174-176 (1994) · Zbl 0959.35504 |
| [18] | Lee, C. T., Multi-Soliton Solutions of the Two-mode KdV (2007), Oxford University: Oxford University Oxford, Ph.D. thesis |
| [19] | Hirota, R.; Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85, 8-9, 407-408 (1981) |
| [20] | Majeed, A.; Rafiq, M. N.; Kamran, M.; Abbas, M.; Inc, M., Analytical solutions of the fifth-order time fractional nonlinear evolution equations by the unified method, Mod. Phys. Lett. B, 36, Article 2150546 pp. (2022) |
| [21] | Saliou, Y.; Abbagari, S.; Houwe, A.; Osman, M. S.; Yamigno, D. S.; Crépin, K. T.; Inc, M., W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations, Mod. Phys. Lett. B, 35, Article 2150468 pp. (2021) |
| [22] | Wazwaz, A. M., Multiple soliton solutions and other exact solutions for a two-mode KdV equation, Math. Methods Appl. Sci., 40, 6, 1277-1283 (2017) |
| [23] | Xiao, Z. J.; Tian, B.; Zhen, H. L.; Chai, J.; Wu, X. Y., Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid, Waves Random Complex Media, 31, 6, 1-4 (2016) |
| [24] | Lee, C. C.; Lee, C. T.; Liu, J. L.; Huang, W. Y., Quasi-solitons of the two-mode Korteweg-de Vries equation, Eur. Phys. J. Appl. Phys., 52, 11-301 (2010) |
| [25] | Hong, W. P.; Jung, Y. D., New non-traveling solitary wave solutions for a second-order Korteweg-de Vries equation, Z. Naturforsch. A, 54, 375-378 (1999) |
| [26] | Zhu; Huang, H. C.; Xue, W. M., Solitary wave solutions having two wave modes of KdV-type and KdV-Burgers-type, Chin. J. Phys., 35, 6, 633-639 (1997) · Zbl 07844742 |
| [27] | Seadawy, A. R., Exact solutions of a two-dimensional nonlinear Schrodinger equation, Appl. Math. Lett., 25, 687-691 (2012) · Zbl 1241.35191 |
| [28] | Seadawy, A. R., Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method, Eur. Phys. J. Plus, 130, 182, 1-10 (2015) |
| [29] | Alamri, S. Z.; Seadawy, A. R.; Al-Sharari, H. M., Study of optical soliton fibers with power law model by means of higher order nonlinear Schrodinger dynamical system, Results Phys., 13, Article 102251 pp. (2019) |
| [30] | Seadawy, A., The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrodinger equation and its solutions, Optik, Int. J. Light Electron Opt., 139, 31-43 (2017) |
| [31] | Seadawy, A. R.; Iqbal, M.; Lu, D., Construction of soliton solutions of the modify unstable nonlinear Schrodinger dynamical equation in fiber optics, Indian J. Phys., 94, 6, 823-832 (2020) |
| [32] | Ali, A.; Seadawy, A.; Lu, D., Soliton solutions of the nonlinear Schrodinger equation with the dual power law nonlinearity and resonant nonlinear Schrodinger equation and their modulation instability analysis, Optik, Int. J. Light Electron Opt., 145, 79-88 (2017) |
| [33] | Arshad, M.; Seadawy, A.; Lu, D., Elliptic function and solitary wave solutions of the higher-order nonlinear Schrodinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, Eur. Phys. J. Plus, 132, 371 (2017) |
| [34] | Ali, M. N.; Seadawy, A. R.; Husnine, S. M.; Tariq, K. U., Optical pulse propagation in monomode fibers with higher order nonlinear Schrodinger equation, Optik, Int. J. Light Electron Opt., 156, 356-364 (2018) |
| [35] | Seadawy, A. R.; Chemaa, N., Improved perturbed nonlinear Schrodinger dynamical equation with type of Kerr law nonlinearity with optical soliton solutions, Phys. Scr., 95, 6, Article 065209 pp. (2020) |
| [36] | Nasreen, N.; Seadawy, A. R.; Lu, D.; Albarakati, W. A., Dispersive solitary wave and soliton solutions of the generalized third order nonlinear Schrodinger dynamical equation by modified analytical method, Results Phys., 15, Article 102641 pp. (2019) |
| [37] | Cao, Y.; Malomed, B. A.; He, J., Two (2+1)-dimensional integrable nonlocal nonlinear Schrodinger equations: breather, rational and semi-rational solutions, Chaos Solitons Fractals, 114, 99-107 (2018) · Zbl 1415.35082 |
| [38] | Zhong, W. P.; Belic, M.; Malomed, B. A.; Huang, Tingwen, Breather management in the derivative nonlinear Schrodinger equation with variable coefficients, Ann. Phys., 355, 313-321 (2015) · Zbl 1343.35221 |
| [39] | Sakaguchi, H.; Malomed, B. A., Singular solitons, Phys. Rev. E, 101, Article 012211 pp. (2020) |
| [40] | Malomed, B. A., A variety of dynamical settings in dual-core nonlinear fibers, (Handbook of Optical Fibers (2018), Springer: Springer Singapore) |
| [41] | Chen, S. S.; Tian, B.; Sun, Y.; Zhang, C. R., Generalized Darboux transformations, rogue waves, and modulation instability for the coherently coupled nonlinear Schrodinger equations in nonlinear optics, Ann. Phys. (Berlin), 531, Article 1900011 pp. (2019) · Zbl 07761446 |
| [42] | Javid, A.; Seadawy, A. R.; Raza, N., Dual-wave of resonant nonlinear Schrödinger’s dynamical equation with different nonlinearities, Phys. Lett. A, 407, Article 127446 pp. (2021) · Zbl 07411280 |
| [43] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Shallow water in an open sea or a wide channel: auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system, Chaos Solitons Fractals, 138, Article 109950 pp. (2020) · Zbl 1490.35314 |
| [44] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations, Appl. Math. Lett., 104, Article 106170 pp. (2020) · Zbl 1437.86001 |
| [45] | Zhang, C. R.; Tian, B.; Qu, Q. X.; Liu, L.; Tian, H. Y., Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber, Z. Angew. Math. Phys., 71, 18 (2020) · Zbl 1508.35164 |
| [46] | Du, X. X.; Tian, B.; Qu, Q. X.; Yuan, Y. Q.; Zhao, X. H., Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fractals, 134, Article 109709 pp. (2020) · Zbl 1483.35177 |
| [47] | Chen, S. S.; Tian, B.; Chai, J.; Wu, X. Y.; Du, Z., Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrodinger equation for the attosecond pulses in the optical fiber communication, Waves Random Complex Media, 30, 389-402 (2020) · Zbl 1498.78032 |
| [48] | Wang, M.; Tian, B.; Sun, Y.; Zhang, Z., Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles, Comput. Math. Appl., 79, 576 (2020) · Zbl 1443.76233 |
| [49] | Chen, Y. Q.; Tian, B.; Qu, Q. X.; Li, H.; Zhao, X. H.; Tian, H. Y.; Wang, M., Ablowitz-Kaup-Newell-Segur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg-de Vries equation in plasma physics, fluid dynamics or atmospheric science, Int. J. Mod. Phys. B, 34, Article 2050226 pp. (2020) · Zbl 1451.35158 |
| [50] | Zhou, Q.; Huang, Z.; Sun, Y.; Triki, H.; Liu, W.; Biswas, A., Collisions of three-solitons in an optical communication system with third-order dispersion and nonlinearity, Nonlinear Dyn., 111, 5757-5765 (2023) |
| [51] | Liu, X.; Triki, H.; Zhou, Q.; Liu, W.; Biswas, A., Analytic study on interactions between periodic solitons with controllable parameters, Nonlinear Dyn., 94, 703-709 (2018) |
| [52] | 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, Article 168673 pp. (2022) |
| [53] | Alquran, M.; Ali, M.; Jaradat, I.; Al-Ali, N., Changes in the physical structures for new versions of the Degasperis-Procesi-Camassa-Holm model, Chin. J. Phys., 71, 85-94 (2021) · Zbl 07835665 |
| [54] | Alquran, M., Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term, Results Phys., 28, Article 104577 pp. (2021) |
| [55] | Wazwaz, A. M., The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations, Appl. Math. Comput., 169, 321-338 (2005) · Zbl 1121.65359 |
| [56] | Kudryashov, N. A., On one of methods for finding exact solutions of nonlinear differential equations (2011) |
| [57] | Ryabov, P. N.; Sinelshchikov, D. I.; Kochanov, M. B., Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Appl. Math. Comput., 218, 3965-3972 (2011) · Zbl 1246.35015 |
| [58] | Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277, 212-218 (2000) · Zbl 1167.35331 |
| [59] | He, J. H.; Wu, X. H., Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30, 3, 700-708 (2006) · Zbl 1141.35448 |
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