Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation

Authors

DOI:

https://doi.org/10.53391/mmnsa.2021.01.003

Keywords:

Eckhaus equation, cubic nonlinear Klein Gordon equation, complex hyperbolic trigonometric travelling wave solutions, non-integer balancing term

Abstract

In this study, an alternative method has been applied to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences. We argue that this method can be used for some special nonlinear partial differential equations (NPDEs) in which the balancing methods are not integer. A number of new complex hyperbolic trigonometric traveling wave solutions have been successfully generated in the Eckhaus equation (EE) and nonlinear Klein-Gordon (nKG) equation models associated with the Schrödinger equation. The graphs representing the stationary wave are presented by giving specific values to the parameters contained in these solutions. Finally, some discussions about new complex solutions are given. It is discussed by giving physical meaning to the constants in traveling wave solutions, which are physically important as well as mathematically. These discussions are supported by three-dimensional simulation. In order to eliminate the complexity of the process and to save time, computer package programs have been utilized.

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References

Alquran, M., Jarrah, A., and Krishnan, E. Solitary Wave Solutions of the Phi-Four Equation and the Breaking Soliton System by Means of Jacobi Elliptic Sine-Cosine Expansion Method. Nonlinear Dynamics and Systems Theory, 18(3), 233-240, (2018).

Durur, H., and Yokuş, A. Discussions on diffraction and the dispersion for traveling wave solutions of the (2+ 1)-dimensional paraxial wave equation. Mathematical Sciences, 1-11, (2021) https://link.springer.com/article/10.1007/s40096-021-00419-z.

Yokuş, A., Durur, H., and Duran, S. Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation. Optical and Quantum Electronics, 53(402), (2021) https://doi.org/10.1007/s11082-021-03036-1.

Malfliet, W. and Hereman, W. The tanh method: II. Perturbation technique for conservative systems. Physica Scripta, 54(6), 569, (1996).

Duran, S., and Karabulut, B. Nematicons in liquid crystals with Kerr Law by sub-equation method. Alexandria Engineering Journal, (2021) https://doi.org/10.1016/j.aej.2021.06.077.

Caudrelier, V.. Interplay between the Inverse Scattering Method and Fokas’s Unified Transform with an Application. Studies in Applied Mathematics, 140(1), 3-26, (2018).

Zhang, Q., Xiong, M., and Chen, L. The First Integral Method for Solving Exact Solutions of Two Higher Order Nonlinear Schrödinger Equations, Journal of Advances in Applied Mathematics, 4(1), (2019).

Feng, Z. and Wang, X. The first integral method to the two-dimensional Burgers–Korteweg–de Vries equation. Physics Letters A, 308(2-3), 173-178, (2003).

Fan, E. Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4-5), 212-218, (2000).

Tariq, H. et al. New travelling wave analytic and residual power series solutions of conformable Caudrey-Dodd-Gibbon Sawada-Kotera equation. Results in Physics, 104591, (2021).

Wazwaz, A.-M. The Hirota’s direct method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Ito seventh-order equation. Applied Mathematics and Computation, 199(1), 133-138, (2008).

Akbulut, A. and Kaplan, M. Auxiliary equation method for time-fractional differential equations with conformable derivative. Computers & Mathematics with Applications, 75(3), 876-882, (2018).

Baskonus, H.M. and Bulut, H. On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media, 25(4), 720-728, (2015).

Wang, M., Li, X., and Zhang, J. The (G0/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423, (2018).

Duran, S. Extractions of travelling wave solutions of (2+ 1)-dimensional Boiti–Leon–Pempinelli system via (G0/G, 1/G)–expansion method. Optical and Quantum Electronics, 53(6), 1-12, (2021).

Duran, S. Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G0/G, 1/G)–Expansion Method. Adıyaman University Journal of Science, 10(2), 585-594, (2020).

Yokuş, A., and Durur, H. (G0/G, 1/G)–expansion method for analytical solutions of Jimbo-Miwa equation. Cumhuriyet Science Journal, 42(1), 88-98, (2021).

Durur, H. Energy-carrying wave simulation of the Lonngren-wave equation in semiconductor materials. International Journal of Modern Physics B, 2150213, (2021).

Ali, K.K., Yilmazer, R., Bulut, H., Aktürk, T., and Osman, M.S. Abundant exact solutions to the strain wave equation in micro-structured solids. Modern Physics Letters B, 2150439, (2021).

Yokuş, A., Durur, H., Nofal, T.A., Abu-Zinadah, H., Tuz, M., and Ahmad, H. Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation. Open Physics, 18(1), 1003-1010, (2020).

Ali, K K., Yilmazer, R., Baskonus, H.M., and Bulut, H. New wave behaviors and stability analysis of the Gilson–Pickering equation in plasma physics. Indian Journal of Physics, 95(5), 1003-1008, (2021).

Ali, K.K., Yilmazer, R., Baskonus, H.M., and Bulut, H. Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves. Physica Scripta, 95(6), 065602, (2020).

Akinyemi, L., Şenol, M., Rezazadeh, H., Ahmad, H., and Wang, H. Abundant optical soliton solutions for an integrable (2+ 1)-dimensional nonlinear conformable Schrödinger system. Results in Physics, 25, 104177, (2021).

Duran, S. Breaking theory of solitary waves for the Riemann wave equation in fluid dynamics. International Journal of Modern Physics B, 2150130, (2021).

Taghizadeh, N., Mirzazadeh, M., and Tascan, F., The first-integral method applied to the Eckhaus equation, Applied Mathematics Letters, 25(5), 798-802, (2012).

Calogero, F. and Eckhaus, W. Nonlinear evolution equations, rescalings, model PDEs and their integrability: II. Inverse Problems, 4(1), 11, (1988).

Calogero, F. and De Lillo, S. Cauchy problems on the semiline and on a finite interval for the Eckhaus equation. Inverse Problems, 4(4), L33, (1988).

Calogero, F. The evolution partial differential equation ut= uxxx+ 3 (uxxu 2+ 3 u 2 xu)+ 3 uxu 4. Journal of mathematical physics, 28(3), 538-555, (1987).

Calogero, F. and De Lillo, S. The Eckhaus PDE i t + xx + 2(| |2)x + | |4 = 0. Inverse Problems, 3(4), 633, (1987).

Calogero, F. and De Lillo, S. The Eckhaus equation in an external potential. Journal of Physics A: Mathematical and General, 25(7), L287, (1992).

Zayed, E.M. and Gepreel, K.A. The modified (G0/G)-expansion method and its applications to construct exact solutions for nonlinear PDEs. WSEAS transactions on mathematics, 10(8), 270-278, (2011).

Peskin, M.E. An introduction to quantum field theory, CRC Press, (2018). [33] Khalatnikov, I.M. An introduction to the theory of superfluidity, CRC Press, (2018).

Wazwaz, A.-M. Compactons, solitons and periodic solutions for some forms of nonlinear Klein–Gordon equations. Chaos, Solitons & Fractals, 2(4), 1005-1013, (2006).

Baskonus, H.M. and Bulut, H. New hyperbolic function solutions for some nonlinear partial differential equation arising in mathematical physics, Entropy, 17(6), 4255-4270, (2015).

Bulut, H. and Başkonus, H.M., On The Geometric Interpretations of The Klein-Gordon Equation And Solution of The Equation by Homotopy Perturbation Method. Applications & Applied Mathematics, 7(2), (2012).

Orion, T. Klein–Gordon equation. Encyclopedia of Mathematics, (2012). https://encyclopediaofmath.org/wiki/Klein–Kordon equation.

Yokus, A. Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method. International Journal of Modern Physics B, 32(29), 1850365, (2018).

Yokus, A. and Kaya, D. Numerical and exact solutions for time fractional Burgers’ equation. Journal of Nonlinear Sciences and Applications, 10(7), 3419-3428, (2017).

Yokus, A. and Kaya, D. Conservation laws and a new expansion method for sixth order Boussinesq equation. in AIP Conference Proceedings, AIP Publishing, (2015).

Arecchi, F. and Bonifacio, R. Theory of optical maser amplifiers. IEEE Journal of Quantum Electronics, 1(4), 169-178, (1965).

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Published

2021-08-24

How to Cite

Yokuş, A. (2021). Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation. Mathematical Modelling and Numerical Simulation With Applications (MMNSA), 1(1), 24–31. https://doi.org/10.53391/mmnsa.2021.01.003

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Research Articles