Exact and approximate solutions for a generalized form of the Schrödinger nonlinear equation

Subject Areas : Statistics

Behzad Ghanbari ¹

1 - Assistant Professor of Applied Mathematics, Department of Basic Sciences, Kermanshah University of Technology, Iran

Received: 2019-04-13 Accepted : 2019-08-13 Published : 2020-12-21

Keywords: محاسبات نمادین, جواب‌های دقیق و عددی, معادله غیرخطی شرودینگر, معادلات با مشتقات جزئی, روش توابع نمایی کسری تعمیم یافته,

Abstract :

In this paper, we consider a generalized form of nonlinear Schrodinger with second-order spatiotemporal dispersion coefficients. The generalized exponential rational function method (GERFM) have been used to obtain some novel exact optical solutions. Also, a new iterative method is successfully examined to numerical solution of the equation. Several numerical simulations are provided to show the behavior of the exact solution, and reveal the efficiently of the numerical results. It is apparent that both employed methods are simple but quite efficient for the extraction of solutions of the problem. Moreover, they are applicable for solving other nonlinear problems arising in mathematics, physics and other branches of engineering. All computations and numerical simulations are carried out with Mathematica. In this paper, we consider a generalized form of nonlinear Schrodinger with second-order spatiotemporal dispersion coefficients. The generalized exponential rational function method (GERFM) have been used to obtain some novel exact optical solutions. Also, a new iterative method is successfully examined to numerical solution of the equation. Several numerical simulations are provided to show the behavior of the exact solution, and reveal the efficiently of the numerical results. It is apparent that both employed methods are simple but quite efficient for the extraction of solutions of the problem. Moreover, they are applicable for solving other nonlinear problems arising in mathematics, physics and other branches of engineering. All computations and numerical simulations are carried out with Mathematica.

References:

[1] Aly Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions, Optik, 139 (2017) 31-43.

 

[2] M. Arshad, Aly Seadawy and Dianchen Lu, Exact Bright-Dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrödinger equation and its stability, Optik, 138 (2017) 40-49.

 

[3] Aly Seadawy and Dianchen Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability, Results in Physics 7 (2017) 43-48.

 

[4] S. Kheybari, M.T. Darvishi, M.S. Hashemi, Numerical simulation for the space-fractional diffusion equations, Applied Mathematics and Computation 348, 57-69

.

[5] M.S. Hashemi, Z Balmeh, On invariant analysis and conservation laws of the time fractional variant Boussinesq and coupled Boussinesq-Burger’s equations, The European Physical Journal Plus 133 (10), 427

 

[6] Asghar Ali, Aly Seadawy and Dianchen Lu, Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis, Optik , 145 (2017) 79-88.

 

[7] M. Arshad, Aly Seadawy and Dianchen Lu, Elliptic function and Solitary Wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, The European Physical Journal Plus, (2017) 132: 371.

[8] M. Arshad and Aly Seadawy, and Dianchen Lu, Modulation stability and optical soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms and its applications, Superlattices and Microstructures 112 (2017) 422-434

 

[9] Muhammad Nasir Ali, Aly R. Seadawy, Syed Muhammad Husnine, Kalim U. Tariq, Optical pulse propagation in monomode fibers with higher order nonlinear Schrödinger equation, Optik, 156 (2018) 356-364.

 

[10] G. Witham, Linear and Nonlinear Waves, Wiley, New York, (1974).

 

[11] G.P. Agrawal Nonlinear Fiber Optics (fifth ed.), Academic Press, New York (2013).

 

[12] J.M. Christian, G.S. McDonald, T.F. Hodgkinson, P. Chamorro-Posada, Wave envelopes with second-order spatiotemporal dispersion. II. Modulational instabilities and dark Kerr solitons, Phys. Rev. A 86 (2012) 023838023839.

 

[13] Kalim Ul-Haq Tariq and Aly Seadawy, Optical soliton solutions of higher order nonlinear Schrödinger equation in monomode fibers and its applications, Optik, 154 (2018) 785-798.

 

[14] Aly Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, Journal of Electromagnetic Waves and Applications, 31(2017) 14: 1353-1362.

 

[15] M. Inc, A. I. Aliyu, A. Yusuf, D. Baleanu, Optical solitons, explicit solutions and modulation instability analysis with second-order spatio-temporal dispersion, Eur. Phys. J. Plus 132 (2017) 528.

[16] B. Ghanbari, M. Inc. A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation, Eur. Phys. J. Plus., (2018), 133, 1-19.

 

[17] M.S. Osman, B. Ghanbari, New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach. Optik 2018, 175, 328-333.

 

[18] B. Ghanbari, N. Raza, An analytical method for soliton solutions of perturbed Schrödinger’s equation with quadratic-cubic nonlinearity. Mod. Phys. Lett. B 2019, 33.

 

[19] M.S. Osman, Behzad Ghanbari and J.A.T. Machado New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with
Kerr law nonlinearity, Eur. Phys. J. Plus (2019) 134: 20.

 

[20] B. Ghanbari, A. Yusuf, M. Inc, D. Baleanu, The new exact solitary wave solutions and stability analysis for the (2 + 1)-dimensional Zakharov–Kuznetsov equation, Advances in difference equations (2019) 2019:49.

 

[21] B. Ghanbari, Abundant soliton solutions for the Hirota-Maccari equation via the generalized exponential rational function method, Modern Physics Letters B (2019),1950106.

 

[22] B. Ghanbari, M. Inc, L. Rada, Solitary wave solutions to the tzitzeica type equations obtained by a new efficient approach, Journal of Applied Analysis and Computation, 9(2) 2019, 1-22.

 

[23] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl.,316 (2006), 753-763.

[24] S. Bhalekar, V. Daftardar-Gejji, Convergence of the new iterative method, Int. J. of Differential Equations, (2011), Article ID 989065.

 

[25] E. Kreyszig, Introductory Fuctional Analysis with Applications, John Wiley & Sons, New York (1978).