We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.
Citation: Mahmoud A. E. Abdelrahman, Wael W. Mohammed, Meshari Alesemi, Sahar Albosaily. The effect of multiplicative noise on the exact solutions of nonlinear Schrödinger equation[J]. AIMS Mathematics, 2021, 6(3): 2970-2980. doi: 10.3934/math.2021180
We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.
[1] |
M. A. E. Abdelrahman, Global solutions for the ultra-relativistic Euler equations, Nonlinear Anal., 155 (2017), 140–162. doi: 10.1016/j.na.2017.01.014
![]() |
[2] |
C. O. Alves, F. Gao, M. Squassina, M. Yang, Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943–3988. doi: 10.1016/j.jde.2017.05.009
![]() |
[3] |
P. I. Naumkin, J. J. Perez, Higher-order derivative nonlinear Schrödinger equation in the critical case, J. Math. Phys., 59 (2018), 021506. doi: 10.1063/1.5008500
![]() |
[4] |
M. A. E. Abdelrahman, Cone-grid scheme for solving hyperbolic systems of conservation laws and one application, Comput. Appl. Math., 37 (2018), 3503–3513. doi: 10.1007/s40314-017-0527-9
![]() |
[5] |
M. A. E. Abdelrahman, G. M. Bahaa, Elementary waves, Riemann problem, Riemann invariants and new conservation laws for the pressure gradient model, Eur. Phys. J. Plus, 134 (2019), 187. doi: 10.1140/epjp/i2019-12580-7
![]() |
[6] |
M. A. E. Abdelrahman, N. F. Abdo, On the nonlinear new wave solutions in unstable dispersive environments, Phys. Scripta, 95 (2020), 045220. doi: 10.1088/1402-4896/ab62d7
![]() |
[7] |
H. G. Abdelwahed, Nonlinearity contributions on critical MKP equation, J. Taibah Univ. Sci., 14 (2020), 777–782. doi: 10.1080/16583655.2020.1774136
![]() |
[8] |
H. G. Abdelwahed, Super electron acoustic propagations in critical plasma density, J. Taibah Univ. Sci., 14 (2020), 1363–1368. doi: 10.1080/16583655.2020.1822653
![]() |
[9] |
M. K. Sharaf, E. K. El-Shewy, M. A. Zahran, Fractional anisotropic diffusion equation in cylindrical brush model, J. Taibah Univ. Sci., 14 (2020), 1416–1420. doi: 10.1080/16583655.2020.1824743
![]() |
[10] |
A. M. Wazwaz, The integrable time-dependent sine-Gordon with multiple optical kink solutions, Optik, 182 (2019), 605–610. doi: 10.1016/j.ijleo.2019.01.018
![]() |
[11] |
M. A. E. Abdelrahman, M. A. Sohaly, On the new wave solutions to the MCH equation, Indian J. Phys., 93 (2019), 903–911. doi: 10.1007/s12648-018-1354-6
![]() |
[12] |
M. Eslami, Trial solution technique to chiral nonlinear Schrödinger's equation in (1 + 2)-dimensions, Nonlinear Dyn., 85 (2016), 813–816. doi: 10.1007/s11071-016-2724-2
![]() |
[13] |
M. Mirzazadeh, M. Eslami, A. Biswas, 1-Soliton solution of KdV equation, Nonlinear Dyn., 80 (2015), 387–396. doi: 10.1007/s11071-014-1876-1
![]() |
[14] |
B. Ghanbari, C. K. Kuo, 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), 134. doi: 10.1140/epjp/i2019-12635-9
![]() |
[15] |
C. K. Kuo, B. Ghanbari, Resonant multi-soliton solutions to new (3 + 1)-dimensional Jimbo-Miwa equations by applying the linear superposition principle, Nonlinear Dyn., 96 (2019), 459–464. doi: 10.1007/s11071-019-04799-9
![]() |
[16] |
M. A. E. Abdelrahman, M. A. Sohaly, The development of the deterministic nonlinear PDEs in particle physics to stochastic case, Results Phys., 9 (2018), 344–350. doi: 10.1016/j.rinp.2018.02.032
![]() |
[17] | M. A. E. Abdelrahman, S. Z. Hassan, M. Inc, The coupled nonlinear Schrödinger-type equations, Mod. Phys. Lett. B, 34 (2020), 2050078. |
[18] | W. W. Mohammed, Amplitude equation with quintic nonlinearities for the generalized Swift-Hohenberg equation with additive degenerate noise, Adv. Differ. Equ., 1 (2016), 84. |
[19] |
W. W. Mohammed, Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain, Chinese Ann. Math. B, 39 (2018), 145–162. doi: 10.1007/s11401-018-1057-5
![]() |
[20] | W. W. Mohammed, Modulation equation for the stochastic Swift–Hohenberg equation with cubic and quintic nonlinearities on the real line, Mathematics, 6 (2020), 1–12. |
[21] |
H. G. Abdelwahed, E. K. El-Shewy, M. A. E. Abdelrahman, R. Sabry, New super waveforms for modified Korteweg-de-Veries-equation, Results Phys., 19 (2020), 103420. doi: 10.1016/j.rinp.2020.103420
![]() |
[22] | N. W. Ashcroft, N. D. Mermin, Solid state physics, New York: Cengage Learning, 1976. |
[23] |
H. T. Chu, Eigen energies and eigen states of conduction electrons in pure bismithunder size and magnetic fields quatizations, J. Phys. Chem. Solids, 50 (1989), 319–324. doi: 10.1016/0022-3697(89)90494-0
![]() |
[24] |
P. I. Kelley, Self-focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005–1008. doi: 10.1103/PhysRevLett.15.1005
![]() |
[25] |
M. Blencowe, Quantum electromechanical systems, Phys. Rep., 395 (2004), 159–222. doi: 10.1016/j.physrep.2003.12.005
![]() |
[26] |
W. Grecksch, H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stoch. Anal. Appl., 29 (2011), 631–653. doi: 10.1080/07362994.2011.581091
![]() |
[27] |
C. H. Bruneau, L. Di Menza, T. Lehner, Numerical resolution of some nonlinear Schrödinger-like equations in plasmas, Numer. Meth. Part. D. E., 15 (1999), 672–696. doi: 10.1002/(SICI)1098-2426(199911)15:6<672::AID-NUM5>3.0.CO;2-J
![]() |
[28] |
V. Barbu, M. Röckner, D. Zhang, Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach, J. Nonlin. Sci., 24 (2014), 383–409. doi: 10.1007/s00332-014-9193-x
![]() |
[29] |
M. A. E. Abdelrahman, W. W. Mohammed, The impact of multiplicative noise on the solution of the Chiral nonlinear Schrödinger equation, Phys. Scripta, 95 (2020), 085222. doi: 10.1088/1402-4896/aba3ac
![]() |
[30] |
S. Albosaily, W. W. Mohammed, M. A. Aiyashi, M. A. E. Abdelrahman, Exact solutions of the (2 + 1)-dimensional stochastic chiral nonlinear Schrödinger equation, Symmetry, 12 (2020), 1874. doi: 10.3390/sym12111840
![]() |
[31] |
A. Debussche, C. Odasso, Ergodicity for a weakly damped stochastic nonlinear Schrödinger equation, J. Evol. Equ., 5 (2005), 317–356. doi: 10.1007/s00028-005-0195-x
![]() |
[32] | G. E. Falkovich, I. Kolokolov, V. Lebedev, S. K. Turitsyn, Statistics of soliton-bearing systems with additive noise, Phys. Rev. E, 63 (2001), 025601. |
[33] |
A. Debussche, L. Di Menzab, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131–154. doi: 10.1016/S0167-2789(01)00379-7
![]() |
[34] | K. Cheung, R. Mosincat, Stochastic nonlinear Schrö dinger equations on tori, Stoch. Partial Differ., 7 (2019), 169–208. |
[35] |
A. De Bouard, A. Debussche, A semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math., 96 (2004), 733–770. doi: 10.1007/s00211-003-0494-5
![]() |
[36] |
J. Cui, J. Hong, Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differ. Equations, 263 (2017), 3687–3713. doi: 10.1016/j.jde.2017.05.002
![]() |
[37] |
J. Cui, J. Hong, Z. Liu, W. Zhou, Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equation, J. Differ. Equations, 266 (2019), 5625–5663. doi: 10.1016/j.jde.2018.10.034
![]() |
[38] | X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 1 (2015), 117–133. |
[39] |
A. M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Model., 40 (2004), 499–508. doi: 10.1016/j.mcm.2003.12.010
![]() |
[40] | A. M. Wazwaz, The sine-cosine method for obtaining solutions with compact and noncompact structures, Appl. Math. Comput., 159 (2004), 559–576 |
[41] |
E. Yusufoglu, A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution equations by using sine-cosine method, Int. J. Comput. Math., 83 (2006), 915–924. doi: 10.1080/00207160601138756
![]() |