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Optical and Quantum Electronics

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01.05.2024

Variational principle for generalized unstable and modify unstable nonlinear Schrödinger dynamical equations and their optical soliton solutions

verfasst von: Aly R. Seadawy, Bayan A. Alsaedi

Erschienen in: Optical and Quantum Electronics | Ausgabe 5/2024

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Abstract

In this paper, we investigate two types of nonlinear Schrödinger equations (NLSE): the unstable NLSE and the modify unstable NLSE. These equations describe the time evolution of disturbances in unstable media. To solve the proposed equations, we employ the variational principle method that involves selecting trial functions based on the Jost function in different forms. Also, these ansatz functions should be continuous at all intervals and may contain single or two nontrivial variational parameters. After that, we use these trial functions to find the functional integral and Lagrangian of the system without any loss. Besides, we use the amplitude ansatz method to explore new soliton solutions. The obtained results include various solitons, such as bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark-bright soliton solutions, and periodic solitary wave solutions. The results will be displayed through different types of graphs, including 2D, 3D, and contour plots, which effectively highlight their outcomes. These solutions have essential applications in the fields of applied science and engineering. Also, they are stable and analytical solutions. The offered techniques can be utilized to solve numerous nonlinear models in mathematical physics and various applied sciences fields.

Vorheriger Artikel Dynamical behavior of the fractional generalized nonlinear Schrödinger equation of third-order

Nächster Artikel A construction of novel soliton solutions to the nonlinear fractional Kairat-II equation through computational simulation

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Ablowitz, M.J., Ablowitz, M.A., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRef

Agrawal, G.P.: Nonlinear fiber optics. In: Nonlinear Science at the Dawn of the 21st Century, pp. 195–211. Springer, Berlin (2000)

Ahmad, J., Mustafa, Z.: Analysis of soliton solutions with different wave configurations to the fractional coupled nonlinear Schrödinger equations and applications. Opt. Quantum Electron. 55 (2023), article number 1228

Ali, A., Ahmad, J., Javed, S.: Exploring the dynamic nature of soliton solutions to the fractional coupled nonlinear Schrödinger model with their sensitivity analysis. Opt. Quantum Electron. 55 (2023), article number 810

Ali, A., Ahmad, J., Javed, S.: Investigating the dynamics of soliton solutions to the fractional coupled nonlinear Schrödinger model with their bifurcation and stability analysis. Opt. Quantum Electron. 55 (2023), article number 829

Ali, K., Seadawy, A.R., Aziz, N., Rizvi, S.T.R.: Soliton solutions to generalized (2+1)-dimensional Hietarinta-type equation and resonant NLSE along with stability analysis. Int. J. Mod. Phys. B 38(01), 2450009 (2024)ADSCrossRef

Aniqa, A., Ahmad, J.: Soliton solution of fractional Sharma–Tasso–Olever equation via an efficient expansion method. Ain Shams Eng. J. 13(1), 101528 (2022)CrossRef

Arbabi, S., Najafi, M.: Exact solitary wave solutions of the complex nonlinear Schrödinger equations. Optik 127(11), 4682–4688 (2016). https://doi.org/10.1016/j.ijleo.2016.02.008ADSCrossRef

Arshad, M., Seadawy, A.R., Lu, D., Wang, J.: Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations. Results Phys. 6, 1136–1145 (2016). https://doi.org/10.1016/j.rinp.2016.11.043ADSCrossRef

Arshad, M., Seadawy, A.R., Jun, W.: Modulation instability analysis of modify unstable nonlinear schrodinger dynamical equation and its optical soliton solutions. Results Phys. 7, 4153–4161 (2017). https://doi.org/10.1016/j.rinp.2017.10.029ADSCrossRef

Arshad, M., Seadawy, A.R., Lu, D., Wang, J.: Optical soliton solutions of unstable nonlinear Schröodinger dynamical equation and stability analysis with applications. Optik 157, 597–605 (2018). https://doi.org/10.1016/j.ijleo.2017.11.129ADSCrossRef

Bona, J.L., Saut, J.: Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves. Chin. Ann. Math. Ser. B 31(6), 793–818 (2010). https://doi.org/10.1007/s11401-010-0617-0CrossRef

Bona, J.L., Ponce, G., Saut, J., Sparber, C.: Dispersive blow-up for nonlinear Schrödinger equations revisited. J. De Math. Pures et Appl. 102(4), 782–811 (2014). https://doi.org/10.1016/j.matpur.2014.02.006MathSciNetCrossRef

Chabchoub, A., Hoffmann, N., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106(20) (2011). https://doi.org/10.1103/physrevlett.106.204502

Dalfovo, F., Giorgini, S., Pitaevskiĭ, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463–512 (1999). https://doi.org/10.1103/revmodphys.71.463ADSCrossRef

Davydov, A.S.: Solitons in Molecular Systems, p. 113. Reidel, Dordrecht (1985)CrossRef

Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. 305(6), 383–392 (2002). https://doi.org/10.1016/s0375-9601(02)01516-5MathSciNetCrossRef

Faridi, W.A., Tipu, G.H., Myrzakulova, Z., Myrzakulov, R., Akinyemi, L.: Formation of optical soliton wave profiles of Shynaray-IIA equation via two improved techniques: a comparative study. Opt. Quantum Electron. 56 (2024), article number 132

Faridi, W.A., Bakar, M.A., Myrzakulova, Z., Myrzakulov, R., Akgül, A., El Din, S.M.: The formation of solitary wave solutions and their propagation for Kuralay equation. Results Phys. 52, 106774 (2023)CrossRef

Faridi, W.A., Bakar, M.A., Akgül, A., El-Rahman, M.A., El Din, S.M.: Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023)CrossRef

Fedele, R., Miele, G., Palumbo, L., Vaccaro, V.G.: Thermal wave model for nonlinear longitudinal dynamics in particle accelerators. Phys. Lett. 179(6), 407–413 (1993). https://doi.org/10.1016/0375-9601(93)90099-lCrossRef

Ghanbari, B., Kuo, C.-K.: New exact wave solutions of the variable-coefficient (1 + 1)-dimensional Benjamin–Bona–Mahony and (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations via the generalized exponential rational function method. Eur. Phys. J. Plus 134(2019), article number 334

Ghanbari, B.: Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method. Mod. Phys. Lett. B 33(09), 1950106 (2019)ADSMathSciNetCrossRef

Ghanbari, B., Akgül, A.: Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95(7), 075201 (2020)ADSCrossRef

Ghanbari, B., Gómez-Aguilar, J.F.: Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. Mod. Phys. Lett. B 33(32), 1950402 (2019)ADSMathSciNetCrossRef

Ghanbari, B., Gómez-Aguilar, J.F.: New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative. Mod. Phys. Lett. B 33(20), 1950235 (2019)ADSCrossRef

Helal, M.A., Seadawy, A.R.: Variational method for the derivative nonlinear Schrödinger equation with computational applications. Phys. Scr. 80, 350–360 (2009)CrossRef

Helal, M.A., Seadawy, A.R.: Variational method for the derivative nonlinear Schrödinger equation with computational applications. Phys. Scr. 80(3), 035004 (2009). https://doi.org/10.1088/0031-8949/80/03/035004ADSCrossRef

Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple interactions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)ADSCrossRef

Hosseini, K., Kumar, D., Kaplan, M., Bejarbaneh, E.Y.: New exact traveling wave solutions of the unstable nonlinear Schrödinger equations. Commun. Theor. Phys. 68(6), 761 (2017). https://doi.org/10.1088/0253-6102/68/6/761ADSMathSciNetCrossRef

Iqbal, M., Lu, D., Seadawy, A.R., Ashraf, M., Albaqawi, H.S., Khan, K.A., Chou, D.: Investigation of solitons structures for nonlinear ionic currents microtubule and Mikhaillov–Novikov–Wang dynamical equations. Opt. Quantum Electron. 56 (2024), article number 361

Javeed, S., Bleanu, D., Waheed, A., Khan, M.S., Affan, H.: Analysis of homotopy Perturbation Method for solving fractional order differential equations. Mathematics 7(1), 40 (2019). https://doi.org/10.3390/math7010040CrossRef

Kaup, D.J., Malomed, B.A.: Variational principle for the Zakharov–Shabat equations. Physica D 84(3–4), 319–328 (1995). https://doi.org/10.1016/0167-2789(95)00057-bADSMathSciNetCrossRef

Khater, A.H., Callebaut, D.K., Helal, M.A., Seadawy, A.R.: Variational method for the nonlinear dynamics of an elliptic magnetic stagnation line. Eur. Phys. J. D 39, 237–245 (2006)ADSCrossRef

Li, B., Chen, Y.: On exact solutions of the nonlinear Schrödinger equations in optical fiber. Chaos Solitons Fract. 21(1), 241–247 (2004). https://doi.org/10.1016/j.chaos.2003.10.029ADSCrossRef

Li, M., Xu, T., Wang, L.: Dynamical behaviors and soliton solutions of a generalized higher-order nonlinear Schrödinger equation in optical fibers. Nonlinear Dyn. 80(3), 1451–1461 (2015). https://doi.org/10.1007/s11071-015-1954-zCrossRef

Lü, X., Zhu, H., Meng, X., Yang, Z., Tian, B.: Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336(2), 1305–1315 (2007). https://doi.org/10.1016/j.jmaa.2007.03.017MathSciNetCrossRef

Lu, D., Seadawy, A.R., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrödinger equations. Optik 140, 136–144 (2017). https://doi.org/10.1016/j.ijleo.2017.04.032ADSCrossRef

Majid, S.Z., Asjad, M.I., Faridi, W.A.: Solitary travelling wave profiles to the nonlinear generalized Calogero–Bogoyavlenskii–Schiff equation and dynamical assessment. Eur Phys. J. Plus 138 (2023), article number 1040

Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992). https://doi.org/10.1119/1.17120ADSMathSciNetCrossRef

Malfliet, W., Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54(6), 563–568 (1996). https://doi.org/10.1088/0031-8949/54/6/003ADSMathSciNetCrossRef

Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan \((\Phi (\xi )/2)\)-expansion method. Optik 127(10), 4222–4245 (2016). https://doi.org/10.1016/j.ijleo.2016.01.078ADSCrossRef

Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150(4), 1079–1085 (1966). https://doi.org/10.1103/physrev.150.1079ADSCrossRef

Pawlik, M., Rowlands, G.: The propagation of solitary waves in piezoelectric semiconductors. J. Phys. C: Solid State Phys. 8(8), 1189–1204 (1975). https://doi.org/10.1088/0022-3719/8/8/022ADSCrossRef

Rani, A., Ashraf, M., Ahmad, J., Ul-Hassan, Q.M. Soliton solutions of the Caudrey–Dodd–Gibbon equation using three expansion methods and applications. Opt. Quantum Electron. 54 (2022), article number 158

Rizvi, S.T.R., Seadawy, A.R., Ahmed, S., Younis, M., Ali, K.: Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation. Chaos Solitons Fract. 151, 111251 (2021)CrossRef

Rizvi, S.T.R., Seadawy, A.R., Kamran Naqvi, S., Ismail, M.: Bifurcation analysis for mixed derivative nonlinear Schrödinger’s equation, \(\alpha \)-helix nonlinear Schrödinger’s equation and Zoomeron model. Opt. Quant. Electron. 56, 452 (2024)ADSCrossRef

Rizvi, S.T.R., Seadawy, A.R., Ahmed, S.: Bell and Kink type, Weierstrass and Jacobi elliptic, multiwave, kinky breather, M-shaped and periodic-kink-cross rational solutions for Einstein’s vacuum field model. Opt. Quant. Electron. 56, 456 (2024)ADSCrossRef

Seadawy, A.R.: New exact solutions for the KdV equation with higher order nonlinearity by using the variational method. Comput. Math. Appl. 62, 3741–3755 (2011)MathSciNetCrossRef

Seadawy, A.R.: Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67, 172–180 (2014)MathSciNetCrossRef

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)

Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron-positron plasma. Physica A: Stat. Mech. Appl. Phys. A 455, 44–51 (2016)ADSMathSciNetCrossRef

Seadawy, A.R., Lu, D.: Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov–Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys. 6, 590–593 (2016). https://doi.org/10.1016/j.rinp.2016.08.023ADSCrossRef

Seadawy, A.R., Lu, D.: Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability. Results Phys. 7, 43–48 (2017). https://doi.org/10.1016/j.rinp.2016.11.038ADSCrossRef

Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsive-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78, 3620–3632 (2019)MathSciNetCrossRef

Seadawy, A.R., Iqbal, M., Lu, D.: Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics. Indian J. Phys. 94(6), 823–832 (2019). https://doi.org/10.1007/s12648-019-01532-5ADSCrossRef

Seadawy, A.R., Ahmad, A., Rizvi, S.T.R., Ahmed, S.: Bifurcation solitons, Y-type, distinct lumps and generalized breather in the thermophoretic motion equation via graphene sheets. Alex. Eng. J. 87, 374–388 (2024)CrossRef

Sindi, C.T., Manafian, J.: Soliton solutions of the quantum Zakharov–Kuznetsov equation which arises in quantum magneto-plasmas. Eur. Phys. J. Plus 132(2) (2017). https://doi.org/10.1140/epjp/i2017-11354-7

Tariq, K.U., Seadawy, A.R.: Bistable bright–dark solitary wave solutions of the (3 + 1)-dimensional Breaking soliton, Boussinesq equation with dual dispersion and modified Korteweg–de Vries–Kadomtsev–Petviashvili equations and their applications. Results Phys. 7, 1143–1149 (2017). https://doi.org/10.1016/j.rinp.2017.03.001ADSCrossRef

Tipu, G.H., Faridi, W.A., Rizk, D., Myrzakulova, Z., Myrzakulov, R., Akinyemi, L.: The optical exact soliton solutions of Shynaray-IIA equation with model expansion approach. Opt. Quantum Electron. 56 (2024), article number 226

Tonti, E.N.Z.O.: Variational formulation for every nonlinear problem. Int. J. Eng. Sci. 22(11–12), 1343–1371 (1984)MathSciNetCrossRef

Wang, K.-J.: Soliton molecules and other diverse wave solutions of the (2+1)-dimensional Boussinesq equation for the shallow water. Eur. Phys. J. Plus 138 (2023), article number 891

Wang, M.: Solitary wave solutions for variant Boussinesq equations. Phys. Lett. 199(3–4), 169–172 (1995). https://doi.org/10.1016/0375-9601(95)00092-hMathSciNetCrossRef

Wang, K.-J.: Dynamics of complexiton, Y-type soliton and interaction solutions to the \((3+1)-\)dimensional Kudryashov–Sinelshchikov equation in liquid with gas bubbles. Results Phys. 54, 107068 (2023)CrossRef

Wang, K.-J.: On the generalized variational principle of the fractal Gardner equation. Fractals 31(09), 2350120 (2023)ADSCrossRef

Wang, K.-J.: Resonant multiple wave, periodic wave and interaction solutions of the new extended (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 111, 16427–16439 (2023)CrossRef

Wang, K.-J.: Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves. Phys. Scr. 99(1), 015223 (2024)ADSCrossRef

Wang, K.-J., Peng, X.: Generalized variational structure of the fractal modified KdV-Zakharov–Kuznetsov equation. Fractals 31(07), 2350084 (2023)ADSCrossRef

Yang, X., Deng, Z., Yi, W.: A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Differ. Equ. 2015(1) (2015). https://doi.org/10.1186/s13662-015-0452-4

Younas, U., Younis, M., Seadawy, R., Rizvi, S.T.E.: Optical solitons and closed form solutions to (3+1)-dimensional resonant Schrodinger equation. Int. J. Mod. Phys. B 34(30), 2050291 (2020)ADSCrossRef

Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35(5), 908–914 (1972)ADS

Zhang, L., Ying, L., Liu, Y.: New solitary wave solutions for two nonlinear evolution equations. Comput. Math. Appl. 67(8), 1595–1606 (2014). https://doi.org/10.1016/j.camwa.2014.02.017MathSciNetCrossRef

Zhao, Q., Wu, L.: Darboux transformation and explicit solutions to the generalized TD equation. Appl. Math. Lett. 67, 1–6 (2017). https://doi.org/10.1016/j.aml.2016.11.012MathSciNetCrossRef

Zhao, H., Han, J., Wang, W., An, H.: Applications of extended hyperbolic Function method for quintic discrete nonlinear Schrödinger equation. Commun. Theor. Phys. 47(3), 474–478 (2007). https://doi.org/10.1088/0253-6102/47/3/020ADSCrossRef

Zhou, Z., Yan, Z.: Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with PT-symmetric harmonic potential via deep learning. Phys. Lett. 387, 127010 (2021). https://doi.org/10.1016/j.physleta.2020.127010CrossRef

Zulfiqar, A., Ahmad, J.: Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method. Results Phys. 19, 103476 (2020). https://doi.org/10.1016/j.rinp.2020.103476CrossRef

Titel: Variational principle for generalized unstable and modify unstable nonlinear Schrödinger dynamical equations and their optical soliton solutions
verfasst von: Aly R. Seadawy
Bayan A. Alsaedi
Publikationsdatum: 01.05.2024
Verlag: Springer US
Erschienen in: Optical and Quantum Electronics / Ausgabe 5/2024
Print ISSN: 0306-8919
Elektronische ISSN: 1572-817X
DOI: https://doi.org/10.1007/s11082-024-06417-4

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