Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of Some Nonlinear Schrödinger’s Equations

Yıl 2020, Cilt: 9 Sayı: 2, 175 - 184, 30.12.2020

İbrahim Enam İnan , Ünal İç

https://doi.org/10.46810/tdfd.733958

Öz

In this study, we found the traveling wave solutions of these equations by applying (3+1)-dimensional nonlinear Schrödinger’s equation and coupled nonlinear Schrödinger’s equation to Generalized Jacobi elliptic function method. We have expressed these solutions both as Jacobi elliptical solutions and trigonometric and hyperbolic solutions. We present two and three dimensional graphics of some solutions we have found. We also state some studies on these equations.

Anahtar Kelimeler

Generalized Jacobi elliptic function method, (3+1)-dimensional nonlinear Schrödinger’s equation, coupled nonlinear Schrödinger’s equation, traveling wave solutions

Bazı Doğrusal Olmayan Schrödinger Denklemlerinin Hareketli Dalga Çözümleri İçin Genelleştirilmiş Jacobi Eliptik Fonksiyon Yöntemi

Öz

Bu çalışmada (3 + 1) boyutlu doğrusal olmayan Schrödinger denklemine ve doğrusal olmayan Schrödinger denklem çiftine Genelleştirilmiş Jacobi eliptik fonksiyon yöntemini uygulayarak bu denklemlerin hareketli dalga çözümlerini bulduk. Bu çözümleri hem Jacobi eliptik çözümler hem de trigonometrik ve hiperbolik çözümler olarak ifade ettik. Bulduğumuz bazı çözümlerin iki ve üç boyutlu grafiklerini sunduk. Ayrıca bu denklemler üzerine yapılan bazı çalışmaları ifade ettik.

Anahtar Kelimeler

Genelleştirilmiş Jacobi eliptik fonksiyon yöntemi, (3 + 1) boyutlu doğrusal olmayan Schrödinger denklemi, doğrusal olmayan Schrödinger denklem çifti, hareketli dalga çözümleri

Kaynakça

• [1] Huai-Tang C, Hong-Qing Z. New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation. Chaos. Solitons and Fractals. 2004;20:765-769.
• [2] Najafi M, Arbabi S. Traveling wave solutions for nonlinear Schrödinger equations. Optik. 2015;126:3992–3997.
• [3] Bulut H, Aksan EN, Kayhan M, Sulaıman TA. New solitary wave structures to the (3+1) dimensional Kadomtsev-Petviashvili and Schrödinger equation. Journal of Ocean Engineering and Science. 2019;4:373-378.
• [4] Arbabi S, Najafi M. Exact solitary wave solutions of the complex nonlinear Schrödinger equations. Optik. 2016;127:4682–4688.
• [5] Bhrawy AH, Abdelkawy MA, Biswas A. Optical solitons in (1+1) and (2+1) dimensions. Optik. 2014;125:1537–1549.
• [6] Esen A, Sulaiman TA, Bulut H, Baskonus HM. Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik. 2018;167:150-156.
• [7] Wazwaz AM. Optical bright and dark soliton solutions for coupled nonlinear Schrödinger (CNLS) equations by the variational iteration method. Optik. 2020;207:164457.
• [8] Manafian J. Optical soliton solutions for Schrödinger type nonlinear evolutionequations by the tan(F(ξ/2)) -expansion method. Optik. 2016;127:4222–4245.
• [9] Bakodah HO, Al Qarni AA, Banaja MA, Zhou Q, Moshokoa SP, Biswas A. Bright and dark Thirring optical solitons with improved adomian decomposition method. Optik. 2017;130:1115–1123.
• [10] Biswas A, Vega-Guzman J, Mahmood MF, Ekici M, Zhou Q, Moshokoa SP, et al. Optical solitons in fiber Bragg gratings with dispersive reflectivity for parabolic law nonlinearity using undetermined coefficients. Optik. 2019;185:39-44.
• [11] Ekici M, Mirzazadeh M, Sonmezoglu A, Zhou Q, Triki H, Ullah MZ, et al. Optical solitons in birefringent fibers with Kerr nonlinearity by exp-function method. Optik. 2017;131:964–976.
• [12] Biswas A, Ekici M, Sonmezoglu A, Belic MR. Optical solitons in fiber Bragg gratings with dispersive reflectivity for quadratic–cubic nonlinearity by extended trial function method. Optik. 2019;185:50-56.
• [13] Biswas A, Aceves AB. Dynamics of solitons in optical fibers. J. Mod. Opt. 2001;48:1135–1150.
• [14] Biswas A, Ekici M, Sonmezoglu A, Belic MR. Highly dispersive optical solitons with Kerr law nonlinearity by F–expansion. Optik. 2019;181:1028–1038.
• [15] Biswas A, Ekici M, Sonmezoglu A, Belic MR. Highly dispersive optical solitons with quadratic–cubic law by F-expansion, Optik. 2019;182:930–943.
• [16] Biswas A, Ekici M, Sonmezoglu A, Belic MR. Highly dispersive optical solitons with Kerr law nonlinearity by extended Jacobi's elliptic function expansion. Optik. 2019;183:395-400.
• [17] Kudryashov NA. General solution of traveling wave reduction for the Kundu–Mukherjee–Naskar model. Optik. 2019;186:22-27.
• [18] Kudryashov NA. Traveling wave solutions of the generalized nonlinear Schrödinger equation with cubic quintic nonlinearity. Optik. 2019;188:27-35.
• [19] Liu W, Zhang Y, Wazwaz AM, Zhou Q. Analytic study on triple-S, triple-triangle structure interactions for solitons in inhomogeneous multi-mode fiber. Applied Mathematics and Computation. 2019;361:325-331.
• [20] Foroutan M, Kumar D, Manafian J, Hoque A. New explicit soliton and other solutions for the conformable fractional Biswas–Milovic equation with Kerr and parabolic nonlinearity through an integration scheme. Optik. 2018;170:190-202.
• [21] Manafian J, Lakestani M. Abundant soliton solutions for the Kundu–Eckhaus equation via tan(F(ξ/2)) -expansion method. Optik. 2016;127:5543–5551.
• [22] Fan E. Two new application of the homogeneous balance method. Phys. Lett. A. 2000;265:353-357.
• [23] Clarkson PA. New similarity solutions for the modified boussinesq equation. J. Phys. A: Math. Gen. 1989;22:2355-2367.
• [24] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992;60:650-654.
• [25] Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A. 2000;277:212-218.
• [26] Fu Z, Liu S, Zhao Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A. 2001;290:72-76.
• [27] Shen S, Pan Z. A note on the Jacobi elliptic function expansion method. Phys. Let. A. 2003;308:143-148.
• [28] Chen Y, Wang Q, Li B. Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations. Z. Naturforsch. A. 2004;59:529-536.
• [29] Chen Y, Yan Z. The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. Chaos Soliton Fract. 2006;29:948-964.