Abstract
This study investigates the (1+1) dimensional dissipative nonlinear Schrödinger equation, which has applications in modeling the evolution of swell in the ocean. Two different and impressive techniques are used for the solutions, which are the Jacobi elliptic function expansion method and the sine-Gorgon expansion method. The suggested techniques are designed to derive the solutions for many other the nonlinear partial differential equations that arising in various branches of sciences. By applying this methods, we obtain novel soliton solutions, which are expressed in terms dark, bright, singular and combo optical solitons. The efficient optical soliton solutions discovered through this methods have the potential to find use in a variety of fields, inclusive electronics, ocean waves, and optical fibers. The modulation instability criterion for governing model is obtained based on the standard linear stability analysis. Additionally, we will also present graphs in 2D and 3D form with constraints conditions.
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References
Ahmad, S., Saifullah, S., Khan, A., Mustafa Inc: New local and nonlocal soliton solutions of a nonlocal reverse space-time mKdV equation using improved Hirota bilinear method. Phys. Lett. A 450, 128393 (2022)
Ahmad, J., Akram, S., Rehman, S.U., Bin Turki, N., Shah, N.A.: Description of soliton and lump solutions to M-truncated stochastic Biswas–Arshed model in optical communication. Results Phys. 51, 106719 (2023a)
Ahmad, J., Akram, S., Noor, K., Nadeem, M., Bucur, A., Alsayaad, Y.: Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber. Sci. Rep. 13, 10877 (2023b)
Ahmad, J., Rani, S., Bin Turki, N., Shah, N.A.: Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method. Results Phys. 52, 106761 (2023c)
Ahmada, J., Mustafa, Z., Rehman, S.U.: Dynamics of exact solutions of nonlinear resonant Schrödinger equation utilizing conformable derivatives and stability analysis. Eur. Phys. J. D 77(123), 1–17 (2023)
Akram, S., Ahmad, J., Rehman, S.U., Sarwar, S., Ali, A.: Dynamics of soliton solutions in optical fibers modelled by perturbed nonlinear Schrödinger equation and stability analysis. Opt. Quantum Electron. 55(450), 1–19 (2023)
Al Qarni, A.A., Bodaqah, A.M., Mohammed, A.S.H.F., Alshaery, A.A., Bakodah, H.O., Biswas, A.: Dark and singular cubic–quartic optical solitons with Lakshmanan–Porsezian–Daniel equation by the improved Adomian decomposition scheme. Ukr. J. Phys. Opt. 24(1), 46–61 (2023)
Ali, K.K., Osman, M.S., Abdel-Aty, M.: New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via sine-Gordon expansion method. Alex. Eng. J. 59(3), 1191–1196 (2020)
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(829), 1–29 (2023a)
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(810), 1–24 (2023b)
Ali Akbar, M., et al.: Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method. Results Phys. 25, 104228 (2021)
Annamalaia, M.: Localization of energy in tubulin system using numerical analysis. Eur. Phys. J. Plus. 137(756), 1–15 (2022)
Arnous, A.H., Biswas, A., Yıldırım, Y., Moraru, L., Aphane, M., Moshokoa, S.P., Alshehri, H.M.: Quiescent optical solitons with Kudryashov’s generalized quintuple-power and nonlocal nonlinearity having nonlinear chromatic dispersion: generalized temporal evolution. Ukr. J. Phys. Opt. 24(2), 105–113 (2023)
Arshed, S., et al.: New soliton solutions of nonlinear Kudryashov’s equation via Improved \(tan(\phi (\eta )/2)\)-expansion approach in optical ber. Kuwait J. Sci. 49(3), 1–16 (2022)
Asghari, Y., Eslami, M., Rezazadeh, H.: Novel optical solitons for the Ablowitz–Ladik lattice equation with conformable derivatives in the optical fibers. Opt. Quantum Electron. 55(930), 1–12 (2023a)
Asghari, Y., Eslami, M., Rezazadeh, H.: Soliton solutions for the time-fractional nonlinear differential-difference equation with conformable derivatives in the ferroelectric materials. Opt. Quantum Electron. 55(289), 1–11 (2023b)
Bakicierler, G., Alfaqeih, S., Misirli, E.: Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations. Rev. Mexic. Fis. 67(6), 060701-1 (2021)
Bayındır, C.: Analytical and numerical aspect of the dissipative nonlinear Shrödinger equation. TWMS J. App. Eng. Math. 6(1), 135–142 (2016)
Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Zahed, H.: Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis. Math. Methods Appl. Sci. 44(5), 4094–4104 (2021)
Cavlak Aslan, E., Gürgöze, L.: Soliton and other function solutions of the potential KdV equation with Jacobi elliptic function method. IJIEA 6(2), 183–188 (2022)
Demiray, H.: An analytical solution to the dissipative nonlinear Schrodinger equation. Appl. Math. Comput. 145, 179–184 (2003)
El-Shiekh, R.M., Gaballah, M.: Solitary wave solutions for the variable-coefficient coupled nonlinear Schrödinger equations and Davey–Stewartson system using modified sine-Gordon equation method. J. Ocean Eng. Sci. 5(2), 180–185 (2020)
Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53, 475–485 (2016)
Ghany, H.A., Zakarya, M.: Exact traveling wave solutions for Wick-type stochastic schamel KdV equation. Phys. Res. Int. 2014, 1–9 (2014)
He, J.-H., Wu, X.-H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006)
Kavitha, L., Muniyappan, A., Prabhu, A., Zdravkovi, S., Jayanthi, S., Gopi, D.: Nano breathers and molecular dynamics simulations in hydrogen-bonded chains. J. Biol. Phys. 39, 15–35 (2013)
Kavitha, L., Parasuraman, E., Muniyappan, A., Gopi, D., Zdravkovi, S.: Localized discrete breather modes in neuronal microtubules. Nonlinear Dyn. 88, 2013–2033 (2017)
Kemaloğlı, B., Yel, G., Bulut, H.: An application of the rational sine-Gordon method to the Hirota equation. Opt. Quantum Electron. 55(658), 1–10 (2022)
Khalil, T.A., Badra, N., Ahmed, H.M., Rabie, W.B.: Optical solitons and other solutions for coupled system of nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index by Jacobi elliptic function expansion method. Optik (Stuttgart) 253(168540), 168540 (2022)
Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Elsevier Science and Technology, Amsterdam (2003)
Kukkar, A., Kumar, S., Malik, S., Biswas, A., Yıldırım, Y., Moshokoa, S.P., Khan, S., Alghamdi, A.A.: Optical solitons for the concatenation model with Kurdryashov’s approaches. Ukr. J. Phys. Opt. 24(2), 155–160 (2023)
Kumar, D., Hosseini, K., Samadani, F.: The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik (Stuttgart) 149, 439–446 (2017)
Kumar, S., Hamid, I., Abdou, M.A.: Dynamic frameworks of optical soliton solutions and soliton-like formations to Schrödinger-Hirota equation with parabolic law non-linearity using a highly efficient approach. Opt. Quantum Electron. 55(1261), 1–31 (2023)
Liang, X., Cai, Z., Wang, M., Zhao, X., Chen, H., Li, C.: Chaotic oppositional sine–cosine method for solving global optimization problems. Eng. Comput. 38(2), 1223–1239 (2022)
Muniyappan, A., et al.: Dark solitons with anti-cubic and generalized anti-cubic nonlinearities in an optical fiber. Optik 255, 168641 (2022a)
Muniyappan, A., et al.: Bright solitons with anti-cubic and generalized anti-cubic nonlinearities in an optical fiber. Optik 254, 168612 (2022b)
Muniyappan, A., Parasuraman, E., Kavitha, L.: Stability analysis and discrete breather dynamics in the microtubulin lattices. Chaos Solitons Fractals 168, 113210 (2023a)
Muniyappana, A., et al.: W-shaped chirp free and chirped bright, dark solitons for perturbed nonlinear Schrödinger equation in nonlinear optical fibers. Proc. Estonian Acad. Sci. 72(2), 128–144 (2023b)
Neirameh, A., Eslami, M.: New optical soliton of stochastic chiral nonlinear Schrödinger equation. Opt. Quantum Electron. 55(444), 1–12 (2023)
Parasuraman, E.: Modulational instability criterion for optical wave propagation in birefringent fiber of Kundu–Eckhaus equation. Optik 243, 167428 (2021a)
Parasuraman, E.: Effect of impurity on modulational instability of localized modes in a discrete quantum ferromagnetic spin chain. J. Magn. Magn. Mater. 539, 168348 (2021b)
Parasuraman, E.: Effect of inter modal dispersion on modulational instability of optical soliton in Kundu–Eckhaus equation with the presence of SPM and XPM. Optik 270, 170020 (2022)
Rabie, W.B., Ahmed, H.M.: Cubic-quartic solitons perturbation with couplers in optical metamaterials having triple-power law nonlinearity using extended F-expansion method. Optik (Stuttgart) 262(169255), 169255 (2022a)
Rabie, W.B., Ahmed, H.M.: Construction cubic-quartic solitons in optical metamaterials for the perturbed twin-core couplers with Kudryashov’s sextic power law using extended F-expansion method. Chaos Solitons Fractals 160(112289), 112289 (2022b)
Rehman, S.U., Seadawy, A.R., Younis, M., Rizvi, S.T.R.: On study of modulation instability and optical soliton solutions: the chiral nonlinear Schrödinger dynamical equation. Opt. Quantum Electron. 53(411), 1–17 (2021)
Rehman, S.U., Ahmad, J., Muhammad, T.: Dynamics of novel exact soliton solutions to Stochastic Chiral Nonlinear Schrödinger Equation. Alex. Eng. J. 79, 568–580 (2023)
Rezazadeh, H., Kumar, D., Sulaiman, T.A., Bulut, H.: New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation. Mod. Phys. Lett. B 33(17), 1950196 (2019)
Seadawy, A.R., Ali, S., Rizvi, S.T.R.: On modulation instability analysis and rogue waves in the presence of external potential: the (n+1)-dimensional nonlinear Schrödinger equation. Chaos Solitons Fractals 161, 112374 (2022a)
Seadawy, A.R., Rizvi, S.T.R., Ahmed, S.: Weierstrass and Jacobi elliptic, bell and kink type, lumps, Ma and Kuznetsov breathers with rogue wave solutions to the dissipative nonlinear Schrödinger equation. Chaos Solitons Fractals 160, 112258 (2022b)
Seadawy, A.R., Rizvi, S.T.R., Ahmed, S.: Solitons collision and multi-peak solutions for a new (3+1)-dimensional NLSE describing pulse propagation in optical fibers. Opt. Quantum Electron. 55(467), 1–24 (2023a)
Seadawy, A.R., Rizvi, S.T.R., Ahmad, A., Ali, K.: Multiwaves, rogue waves, breathers and lump solutions for an NLSE under the influence of self-steeping and Raman effects, along with cubic quintic septimal parameters. Opt. Quantum Electron. 55(771), 1–22 (2023b)
Ulutas, E.: Travelling wave and optical soliton solutions of the Wick-type stochastic NLSE with conformable derivatives. Chaos Solitons Fractals 148, 111052 (2021)
Yazgan, T., Yel, G., Celik, E., Bulut, H.: On survey of the some wave solutions of the non-linear Schrödinger equation (NLSE) in infinite water depth. Gazi J. Sci. 36(2), 819–843 (2023)
Yel, G., Bulut, H.: New wave approach to the conformable resonant nonlinear Schödinger’s equation with Kerr-law nonlinearity. Opt. Quantum Electron. 54(4), 252 (2022)
Yel, G., Kayhan, M., Ciancio, A.: A new analytical approach to the (1+1) dimensional conformable Fisher equation. Math. Model. Numer. Simul. 4(2), 211–220 (2022)
Zaug, C.R., Carter, J.D.: Dissipative models of swell propagation across the Pacific. Stud. Appl. Math. 147, 1519–1537 (2021)
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This study is related to the PhD thesis of the second author.
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Aslan, E.C., Deniz, D. & Inc, M. Analysing of different wave structures to the dissipative NLS equation and modulation instability. Opt Quant Electron 56, 254 (2024). https://doi.org/10.1007/s11082-023-06035-6
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DOI: https://doi.org/10.1007/s11082-023-06035-6