On Survey of the Some Wave Solutions of the Non-Linear Schrödinger Equation (NLSE) in Infinite Water Depth

Abstract

In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.   

Keywords

• The sine-gordon expansion technique
• The modified exp -expansion function technique
• Conformable derivative
• Non-linear schrödinger equation (NLSE)

References

1. [1] Ira Moxley III, F., “Genealized Finite-Difference Time-Domain Schemes for Solving Nonlinear Schrödinger Equations”, Phd. Thesis, (2013).
2. [2] Ablowitz, M.J., Musslimani, Z.H., “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation”, Nonlinearity, 29, 915, (2016).
3. [3] Kumar, D., Manafian, J., Hawlader, F., Ranjbaran, A., “New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method”, Optik, 160: 159-167, (2018).
4. [4] Dusunceli, F., Celik, E., Askin, M., Bulut, H., “New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method”, Indian Journal of Physics, 95(2): 309-314, (2021).
5. [5] Yel, G., “On the new travelling wave solution of a neural communication model”, BAUN Fen Bilimleri Enstitüsü Dergisi, 21(2): 666-678, (2019).
6. [6] Kocak, Z.F., Bulut, H., Yel, G., “The solution of fractional wave equation by using modified trial equation method and homotopy analysis method”, AIP Conference Proceedings, 1637: 504–512, (2014).
7. [7] Biswas, A., Kara, A.H., “1-soliton solution and conservation laws of the generalized dullingottwald-holm equation”, Applied Mathematics and Computation, 217(2): 929-932, (2010).
8. [8] Demiray, S.T., Bulut, H., Celik, E., “Soliton solutions of Wu-Zhang system by generalized Kudryashov method”, AIP Conference Proceedings, 2037(1): (2018).

9. [9] Biswas, A., Moosaei, H., Eslami, M., Mirzazadeh, M., Zhou, Q., Bhrawy, A.H., “Optical soliton perturbation with extended tanh function method”, Optoelectronics and Advanced Materials Rapid Communications, 8(11): 1029-1034, (2014).
10. [10] Bosco, G., Carena, A., Curri, V., Gaudino, R., Poggiolini, P., Bendedetto, S., “Suppression of spurious tones induced by the split-step method in fiber systems simulation”, IEEE Photonics Technology Letters. 12: 489-491, (2000).
11. [11] Chang, Q., Jia, E., Suny, W., “Difference schemes for solving the generalized nonlinear Schrodinger equation”, Journal of Computational Physics, 148: 397-415, (1999).
12. [12] Al-Ghafri, K.S, Rezazadeh, H., “Solitons and other solutions of (3+1)-dimensional space-time fractional modified KdV-Zakharov–Kuznetsov equation”, Applied Mathematics Nonlinear Sciences, 4(2): 289–304, (2019).
13. [13] Jiang, C., Cai, W., Wang, Y., “Optimal error estimate of a conformal Fourier pseudo‐spectral method for the damped nonlinear Schrödinger equation”, Numerical Methods for Partial Differential Equations, 34(4): 1422-1454, (2018).
14. [14] Tariq, K.U., Younis, M., Rizvi, S.T.R., Bulut, H., “M-truncated fractional optical solitons and other periodic wave structures with Schrödinger–Hirota equation”, Modern Physics Letters B, 34: (2020).
15. [15] Li, Y.X., Celik, E., Guirao, J.L.G., Saeed, T., Baskonus, H.M., “On the modulation instability analysis and deeper properties of the cubic nonlinear Schrödinger’s equation with repulsive δ-potential”, Results in Physics, 25: 104303, (2021).
16. [16] Rezazadeh, H., Odabasi, M., Tariq, K.U., Abazari, R., Baskonus, H. M., “On the conformable nonlinear Schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients”, Chinese Journal of Physics, (2021). DOI: doi.org/10.1016/j.cjph.2021.01.012 [17] Gao, W., Ismael, H.F., Husien, A.M., Bulut, H., Baskonus, H.M., “Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law”, Applied Sciences, 10(1): (2020).
17. [18] Gao, W., Jhangeer, A., Baskonus, H.M., Yel, G., “New exact solitary wave solutions, bifurcation analysis and rst order conserved quantities of resonance nonlinear Shrödinger’s equation with Kerr law nonlinearity”, Authorea, (2020).
18. [19] Karjanto, N., “The nonlinear Schrödinger equation: A mathematical model with its wide-ranging applications”, Pattern Formation and Solitons, arXiv:1912.10683v1, (2019).
19. [20] Debnath, L., “Nonlinear Partial Differential Equations for Scientists and Engineers”, 3rd Edition, Springer, (2012).
20. [21] Chabchoub, A., Hoffmann, N., Onorato, M., and Akhmediev N., “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves”, Physical Review X 2, 011015, (2012).
21. [22] Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F., “Rogue waves and their generating mechanisms in different physical contexts”, Physics Reports, 528: 47 – 89, (2013).
22. [23] Zakharov, V. E., “Stability of Periodic Waves of Finite Amplitude on a Surface of Deep Fluid”, Journal of Applied Mechanics and Technical, Physics 2, 190, (1968).
23. [24] Yuen, H. C., and Lake, B. M., “Nonlinear Deep Water Waves: Theory and Experiment”, Physics of Fluids, 18: 956, (1975).
24. [25] Yuen, H. C., and Lake, B. M., “Nonlinear Dynamics of Deep-Water Gravity Waves”, Advances in Applied Mechanics, 22: 67, (1982).
25. [26] Khalila, R., Horania, M. A., Yousefa, A., and Sababheh, M., “A New Definition of Fractional Derivative”, Journal of Computational and Applied Mathematics, 264: 65-70, (2014).
26. [27] Atangana, A., Baleanu, D., and Alsaedi, “A New properties of conformable derivative”, Open Mathematics, 13: 889-898, (2015).
27. [28] Yel, G., “New wave patterns to the doubly dispersive equation in nonlinear dynamic elasticity”, Pramana – Journal of Physics, 94(1): 79, (2020).
28. [29] Kumar, A., Ilhan, E., Ciancio, A., Yel, G., Baskonus, H.M., “Extractions of some new travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation”, AIMS Mathematic, 6(5): 4238-4264, (2021).
29. [30] Yan, C., “A simple transformation for nonlinear waves”, Physics Letters A, 224: 77–84. 45, (1996).
30. [31] Yan Z., Zhang, H., “New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics”, Physics Letters A, 252: 291–296, 46, (1999).
31. [32] Chong, Y. D., “MH2801: Complex Methods for the Sciences”, Nanyang Technological University, (2016). Available from: http://www1.spms.ntu.edu.sg/~ydchong/teaching.html
32. [33] Hafez, M.G., Alam M.N., and Akbar M.A., “Application of the exp(-Φ(η))-expansion Method to Find Exact Solutions for the Solitary Wave Equation in an Unmagnatized Dusty Plasma”, World Applied Sciences Journal, 32(10): 2150-2155, (2014).
33. [34] Roshid, H.O. and Rahman, M.A., “The exp(−Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations”, Results in Physics, 4(150): 150-155, (2014).
34. [35] Shemer, L., Kit, E. and Jiao, H., “An experimental and numerical study of the spatial evolution of unidirectional nonlinear water-wave groups”, Physics of Fluids, 14(10): 3380, (2002).