Abstract
This study investigates a stochastic nonlinear Schrödinger equation (1+1) dimensional with a random potential. The equation under consideration is crucial in the study of the evolution of nonlinear dispersive waves in a completely disordered medium. By employing the \({\phi} ^{6}\) model expansion technique, we will derive stochastic exact solutions for this stochastic partial differential equation. The obtained solutions can be expressed as exponential type and the results show that this method is effective and simple for solving such equations. Additionally, the study of nonlinear equations in a random environment is important as solitons are known to be stable against mutual collisions and behave like particles. However, it is difficult to describe realistic physical phenomena with variable or constant coefficient nonlinear equations.
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A beam that narrows down to itself in nonlinear medium.
References
Abdusalam, H.A.: On an improved complex tanh-function method. Int. J. Nonlin. Sci. Numer. Simul. 6(2), 99–106 (2005)
Alkhidhr, H.A., Abdelrahman, M.A.: Wave structures to the three coupled nonlinear Maccaris systems in plasma physics. Res. Phys. 33, 105092 (2021)
Alkhidhr, H.A., Abdelwahed, H.G., Abdelrahman, M.A., Alghanim, S.: Some solutions for a stochastic NLSE in the unstable and higher order dispersive environments. Res. Phys. 34, 105242 (2022)
Aminikhah, H., Moosaei, H., Hajipour, M.: Exact solutions for nonlinear partial differential equations via Exp-function method. Numer. Meth. Partial Diff. Eq. 26(6), 1427–1433 (2010)
Batool, F., Akram, G.: Application of extended Fan sub-equation method to (1+ 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution. Opt. Quant. Electr. 49(11), 1–9 (2017)
Belaouar, R., De Bouard, A., Debussche, A.: Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion. Stoch. Part. Diff. Eq.: Anal. Comput. 3(1), 103–132 (2015)
Bibi, K.: The \(\phi ^6\)-model expansion method for solving the Radhakrishnan-Kundu-Lakshmanan equation with Kerr law nonlinearity. Optik 234, 166614 (2021)
Cheemaa, N., Younis, M.: New and more exact traveling wave solutions to integrable (2+ 1)-dimensional Maccari system. Nonlin. Dyn. 83(3), 1395–1401 (2016)
Corwin, I., Shen, H.: Some recent progress in singular stochastic partial differential equations. Bull. Am. Math. Soci. 57(3), 409–454 (2020)
De Bouard, A., Debussche, A.: The nonlinear Schrödinger equation with white noise dispersion. J. Funct. Anal. 259(5), 1300–1321 (2010)
Debussche, A., Weber, H.: The Schrödinger equation with spatial white noise potential (2018)
El-Wakil, S.A., Abdou, M.A.: The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations. Chaos, Solit. & Fract. 36(2), 343–353 (2008)
Iqbal, M.S.: Solutions of boundary value problems for nonlinear partial differential equations by fixed point methods. T.U, Graz library (2011)
Jiong, S.: Auxiliary equation method for solving nonlinear partial differential equations. Phys. Lett. A 309(5–6), 387–396 (2003)
Marty, R.: On a splitting scheme for the nonlinear Schrödinger equation in a random medium (2006)
Mohammed, W.W.: Modulation equation for the stochastic Swift Hohenberg equation with cubic and quintic nonlinearities on the real line. Mathematics 7(12), 1217 (2019)
Mohammed, W.W.: Approximate solutions for stochastic time-fractional reaction diffusion equations with multiplicative noise. Math. Meth. Appl. Sci. 44(2), 2140–2157 (2021)
Mohammed, W.W., Ahmad, H., Hamza, A.E., Aly, E.S., El-Morshedy, M., Elabbasy, E.M.: The exact solutions of the stochastic GinzburgLandau equation. Res. Phys. 23, 103988 (2021)
Mohammed, W.W., Iqbal, N., Botmart, T.: Additive noise effects on the stabilization of fractional-space diffusion equation solutions. Mathematics 10(1), 130 (2022)
Paredes, A., Olivieri, D.N., Michinel, H.: From optics to dark matter: a review on nonlinear Schrödinger-Poisson systems. Physica D: Nonlin. Phenom. 403, 132301 (2020)
Prévôt, C., Röckner, M.: A concise course on stochastic partial differential equations (Vol. 1905, pp. vi+-144). Berlin: Springer (2007)
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. Comp. 218(7), 3965–3972 (2011)
Vitanov, N.K.: On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation. Commun. Nonlin. Sci. Numer. Simul. 16(11), 4215–4231 (2011)
Wazwaz, A.M.: The Hirota s direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl. Math. Comput. 201(1–2), 489–503 (2008)
Yasar, E.: New travelling wave solutions to the Ostrovsky equation. Appl. Math. Comp. 216(11), 3191–3194 (2010)
Yomba, E.: The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations. Phys. Lett. A 336(6), 463–476 (2005)
Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.T.R.: Dispersive of propagation wave structures to the Dullin–Gottwald–Holm dynamical equation in a shallow water waves. Chinese J. Phys. 68, 348–364 (2020)
Younis, M., Sulaiman, T.A., Bilal, M., Rehman, S.U., Younas, U.: Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation. Commun. Theoret. Phys. 72(6), 065001 (2020)
Younis, M., Seadawy, A.R., Baber, M.Z., Husain, S., Iqbal, M.S., Rizvi, S.T.R., Baleanu, D.: Analytical optical soliton solutions of the Schrödinger-Poisson dynamical system. Res. Phys. 27, 104369 (2021)
Zayed, E.M., Al-Nowehy, A.G.: Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques. Optik 143, 84–103 (2017)
Zayed, E.M., Al-Nowehy, A.G.: New generalized \(\phi ^6\)- model expansion method and its applications to the \((3+ 1)\) dimensional resonant nonlinear Schrödinger equation with parabolic law nonlinearity. Optik 214, 164702 (2020)
Zhao, Y.M.: F-expansion method and its application for finding new exact solutions to the Kudryashov–Sinelshchikov equation. J. Appl. Math. (2013). https://doi.org/10.1155/2013/895760
Zhou, Q., Yao, D., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic-law nonlinearities. J. Modern Optics 60(19), 1652–1657 (2013)
Zhou, Q., Xiong, X., Zhu, Q., Liu, Y., Yu, H., Yao, P., Belicd, M.: Optical solitons with nonlinear dispersion in polynomial law medium. J. Optoelectr. Adv. Mater. 17, 82–86 (2015)
Ziane, D., Hamdi Cherif, M., Baleanu, D., Belghaba, K.: Non-differentiable solution of nonlinear biological population model on cantor sets. Fractal Fract. 4(1), 5 (2020)
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Hussain, S., Iqbal, M.S., Ashraf, R. et al. Exploring nonlinear dispersive waves in a disordered medium: an analysis using \(\phi ^6\) model expansion method. Opt Quant Electron 55, 651 (2023). https://doi.org/10.1007/s11082-023-04851-4
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DOI: https://doi.org/10.1007/s11082-023-04851-4