New Exact Solutions of Cubic Nonlinear Schrödinger Equation by Using Extended Trial Equation Method

Yıl 2020, , 582 - 588, 25.09.2020

Yusuf Pandır , Tural Ağır

https://doi.org/10.35414/akufemubid.709937

Cited By: 2

Öz

In this study, the extended trial equation method, which allows to obtain exact solutions of the partial differential equations, is investigated. This proposed method is applied to the cubic nonlinear Schrödinger equation and different new exact solutions are obtained. We can state that these new exact solutions are new exact solutions that are not find in the literature. In addition, two and three dimensional graphics drawn to show the physical behavior of these new exact solutions. 

In this study, the extended trial equation method, which allows to obtain exact solutions of the partial differential equations, is investigated. This proposed method is applied to the cubic nonlinear Schrödinger equation and different new exact solutions are obtained. We can state that these new exact solutions are new exact solutions that are not find in the literature. In addition, two and three dimensional graphics drawn to show the physical behavior of these new exact solutions. 

Anahtar Kelimeler

Extended Trial Equation Method, Nonlinear Partial Differential Equations, Schrödinger Equation with a Cubic Nonlinearity, Exact Solutions

Genişletilmiş Deneme Denklemi Yöntemi ile Kübik Lineer Olmayan Schrödinger Denkleminin Yeni Tam Çözümleri

Öz

Bu çalışmada, kısmi türevli diferansiyel denklemlerin tam çözümlerinin elde edilmesine olanak sağlayan genişletilmiş deneme denklem yöntemi incelenmiştir. Önerilen bu yöntem kübik lineer olmayan Schrödinger denklemine uygulanmış ve farklı yeni tam çözümleri elde edilmiştir. Elde edilen bu yeni tam çözümlerin literatürde bulunmayan yeni tam çözümleri olduğunu ifade edebiliriz. Ayrıca, bulunan bu yeni tam çözümlerin fiziksel davranışlarını göstermek için iki ve üç boyutlu grafikleri çizilmiştir.

Bu çalışmada, kısmi türevli diferansiyel denklemlerin tam çözümlerinin elde edilmesine olanak sağlayan genişletilmiş deneme denklem yöntemi incelenmiştir. Önerilen bu yöntem kübik lineer olmayan Schrödinger denklemine uygulanmış ve farklı yeni tam çözümleri elde edilmiştir. Elde edilen bu yeni tam çözümlerin literatürde bulunmayan yeni tam çözümleri olduğunu ifade edebiliriz. Ayrıca, bulunan bu yeni tam çözümlerin fiziksel davranışlarını göstermek için iki ve üç boyutlu grafikleri çizilmiştir.

Anahtar Kelimeler

Genişletilmiş Deneme Denklem Yöntemi, Kısmi Türevli Diferansiyel Denklemler, Kübik Lineer Olmayan Schrödinger Denklemi, Tam Çözümler

Kaynakça

• Ablowitz, M.J., Prinari, B. and Trubatch, A.D., 2004. Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge Univ. Press.
• Akbar, M.A., Ali, N.H.M. and Mohyud-Din, S.T., 2013. The modified alternative -expansion method to nonlinear evolution equation: application to the (1+1)-dimensional Drinfel’d-Sokolov-Wilson equation. SpringerPlus, 327, 2-16.
• Chand, F. and Malik, A.K., 2012. Exact traveling wave solutions of some nonlinear equations using -expansion method. International Journal of Nonlinear Science, 14(4), 416-424.
• Gurefe, Y., Sonmezoglu A. and Misirli, E., 2011. Application of trial equation method to the nonlinear partial differential equations arising in mathematical physics. Pramana-Journal of Physics, 77(6), 1023-1029.
• Gurefe, Y., Sonmezoglu A. and Misirli, E., 2012. Application of an irrational trial equation method to high dimensional nonlinear evolution equations. Journal of Advanced Mathematical Studies, 5(1), 41-47.
• Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M., 2013. Extended trial equation method to generalized nonlinear partial differential equations. Applied Mathematics and Computation, 219(10), 5253-5260.
• Hietarinta, J., 1997. Hirota's bilinear method and its generalization. International Journal of Modern Physics A, 12(1), 43-51.
• Imanli, M.I., 2006. Nonlinear Schrödinger equations in homogenous space. MSc thesis, Fırat University, Elazig.
• Kaplan, M., Ünsal, Ö. and Bekir, A., 2016. Exact solutions of nonlinear Schrödinger equation by using symbolic computation. Mathematical Methods in the Applied Science, 39, 2093-2099.
• Liu, C.S., 2006. Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications. Communications in Theoretical Physics, 45(2), 219-223.
• Liu, C.S., 2010. Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Computer Physics Communications, 181(2), 317-324.
• Pandir, Y., Gurefe, Y., Kadak, U. and Misirli, E., 2012. Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution. Abstract and Applied Analysis, 2012, Article ID 478531, 16 pp.
• Pandir, Y., Gurefe, Y. and Misirli, E. 2013. Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation. Physica Scripta, 87(2), 025003, 12 pp.
• Pandir Y., 2014. Symmetric Fibonacci function solutions of some nonlinear partial differential equations. Applied Mathematics & Information Science, 8, 2237-2241.
• Pashaev, O. and Tanoglu, G., 2005. Vector shock soliton and the Hirota bilinear method. Chaos, Solitons & Fractals, 26, 95-105.
• Shakeel, M. and Mo hyud-Din, S. T., 2015. New G'/G-expansion method and its application to the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK–BBM) equation. Journal of the Association of Arab Universities for Basic & Applied Science, 18(1), 66-81.
• Sulem C. and Sulem P.L. 1999. The nonlinear Schrödinger equation self-focusing and wave collapse. Springer, New-York.
• Tandogan, Y.A., Pandir, Y. and Gurefe, Y., 2013. Solutions of the nonlinear differential equations by use of modified Kudryashov method. Turkish Journal of Mathematics and Computer Science, 1, 54-60.
• Wang M. L., 1996. Exact solutions for compound KdV-Burgers equations. Physics Letters A, 213, 279-287, 1996.
• Wazwaz A. M., 2008. A sine-cosine method for handling nonlinear wave equations. Mathematical and Computer Modellling, 40(5-6), 499-508.