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Exact Solutions of the Nonlinear Schrödinger Equation With Anti Cubic Nonlinearity by the exp(-Φ(ξ)) Method

Year 2022, Volume: 22 Issue: 1, 85 - 91, 28.02.2022
https://doi.org/10.35414/akufemubid.1023095

Abstract

This work is devoted to obtain exact solutions of the nonlinear Schrödinger equation with anti cubic nonlinearity by method. This equation plays a crucial role in nonlinear optics and mathematical physics. The method is an efficient and useful method to find different types of analytical solutions of nonlinear partial differential equations and fractional differential equations. We have used the Maple packet program for the calculations and verification of the solutions for this work.

References

  • Abdou, M.A. 2008. Further improved F-expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dynamics, 52, 277-288.
  • Ablowitz, M.J. and Segur, H., 1981. Solitons and Inverse Scattering Transformation, 4, SIAM, Philadelphia, 1-84.
  • Adem, A.R. and Khalique, C.M., 2016. Conserved quantities and solutions of a (2+1)-dimensional Haragus-Courcelle-Il'ichev model. Computers and Mathematics with Applications, 71, 1129-1136.
  • Akter, J. and Akbar, M.A., 2015. Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in Physics, 5, 125-130.
  • Ali, S. , Rizvia and S.T.R., Younis, M., 2015. Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients. Nonlinear Dynamics, 82, 1755-1762.
  • Alquran, M.T. 2012. Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Appl. Math. Inf. Sci., 6(1), 85-88.
  • Baskonus, H.M., Bulut, H., and Atangana, A. 2016. On the complex hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Material and Structures, 25, 035022.
  • Biswas, A., Jawad, A.J.M. and Zhou, Q., 2018. Resonant optical solitons with anti-cubic nonlinearity. Optik, 157, 525-531.
  • Biswas, A. and Khalique, C.M., 2011. Stationary solutions for nonlinear dispersive Schrödinger's equation. Nonlinear Dynamics, 63, 623-626.
  • Biswas, A. and Konar, S., 2006. Introduction to non-Kerr law optical solitons, 1, CRC Press, Boca Raton FL, 27-54.
  • Biswas, A. and Khalique, C.M., 2011. Stationary solutions for nonlinear dispersive Schrödinger's equation. Nonlinear Dynamics, 63, 623-626.
  • Fan, E., 2000. Extented tanh-function method and its applications to nonlinear equations. Physics Letters A, 277, 212-218.
  • He, J.H. and Abdou, M.A., 2007. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons and Fractals, 34, 1421-1429.
  • Inan, I.E., Ugurlu, Y. and Inc, M., 2015. New Applications of the (G′/G,1/G)-Expansion Method. Acta Physica Polonica A, 128(3), 245-251.
  • Islam, Md. S., Khan, K. and Akbar, M.A., 2015. An analytical method for finding exact solutions of modified Korteweg-de Vries equation. Results in Physics, 5, 131-135.
  • Ismael, H.F., Bulut, H., Baskonus, H.M., Gao, W. 2020. Newly modified method and its application to thr coupled Boussinesq equation in ocean engineering with its linear stability analysis. Communications in Theoritical Physics, 72 (11), 115002.
  • Jawad, A.J., Mirzazadeh, M., Zhou, Q. and Biswas, A., 2017. Optical solitons with anti-cubic nonlinearity using three integration schemes. Superlattices and Microstructures, 105, 1-10.
  • Kaabar, M.K.A., Kaplan, M. and Siri, Z., 2021. New Exact Soliton Solutions of the (3+1)-Dimensional Conformable Wazwaz-Benjamin-Bona-Mahony Equation via Two Novel Techniques. Journal of Function Spaces, 465990.
  • Kaplan, M., Hosseini, K., Samadani, F., Raza, N. 2018. Optical soliton solutions of the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term. Journal of Modern Optics, 65(12) 1431-1436.
  • Kumar, D. and Kaplan, M., 2018. New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques. Chinese Journal of Physics, 56 (5), 2173-2185.
  • Kumar, D., Hosseini, and K., Samadani, F., 2017. The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, 439-446.
  • Lü, X., Tian, B., Zhang, H-Q., Xu, T. and Li, H., 2012. Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms. Nonlinear Dynamics, 67, 2279-2290.
  • Ma, W.X., Abdeljabbar, A. and Asaad, M.G., 2011. Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Applied Mathematics and Computation, 217, 10016-10023.
  • Ma, W.X.A and Lee, J.H., 2009. A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation. Chaos, Solitons and Fractals, 42, 1356-1363.
  • Mirzazadeh, M., Arnous, A.H., Mahmood, M.F., Zerrad, E. and Biswas, A., 2015. Soliton solutions to resonant nonlinear Schrödinger's equation with time-dependent coefficients by trial solution approach. Nonlinear Dynamics, 81, 277-282.
  • Roshid, H.O., Kabir, R.C., Bhowmik, R.C. and Datta, B.K., 2014. Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and exp(-Φ(ξ)) method. SpringerPlus, 3, 692.
  • Wang, M.L., 1995. Solitary wave solutions for variant Boussinesq equations. Physics Letters A, 199, 169-172.
  • Wazwaz, A.M. 2007. Multiple-soliton solutions for the Boussinesq equation. Applied Mathematics and Computation, 192 (2), 479-486.
  • Wazwaz, A.M.2004. The tanh method for travelling wave solutions of nonlinear equations. Applied Mathematics and Computation, 154 (3) ,713-723.
  • Yel, G. Baskonus, H.M. 2019. Solitons in conformable time-fractional Wu-Zhang system arising in coastal design. Pramana 92, 57.
  • Zayed, E.M.E., Alngar, M.E.M. and Al-Nowehy, A.G., 2019. On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms. Optik, 178 488-508.

Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri

Year 2022, Volume: 22 Issue: 1, 85 - 91, 28.02.2022
https://doi.org/10.35414/akufemubid.1023095

Abstract

Bu çalışmada, lineer olmayan kübik-kuintik Schrödinger denkleminin yeni tam çözümleri, üstel -Φ(ξ) yöntemiyle elde edilmiştir. Bu denklem, lineer olmayan optikte ve matematiksel fizikte büyük bir öneme sahiptir. Üstel -Φ(ξ) yöntemi, lineer olmayan kısmi diferensiyel denklemler ve kesir mertebeden kısmi diferensiyel denklemlerin farklı tipte analitik çözümlerini bulmada kullanılan oldukça elverişli ve kullanışlı bir metottur. Bu çalışmada yapılan hesaplamalarda ve çözümlerin doğruluğunun teyit edilmesinde Maple paket programı kullanılmıştır.

References

  • Abdou, M.A. 2008. Further improved F-expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dynamics, 52, 277-288.
  • Ablowitz, M.J. and Segur, H., 1981. Solitons and Inverse Scattering Transformation, 4, SIAM, Philadelphia, 1-84.
  • Adem, A.R. and Khalique, C.M., 2016. Conserved quantities and solutions of a (2+1)-dimensional Haragus-Courcelle-Il'ichev model. Computers and Mathematics with Applications, 71, 1129-1136.
  • Akter, J. and Akbar, M.A., 2015. Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in Physics, 5, 125-130.
  • Ali, S. , Rizvia and S.T.R., Younis, M., 2015. Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients. Nonlinear Dynamics, 82, 1755-1762.
  • Alquran, M.T. 2012. Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Appl. Math. Inf. Sci., 6(1), 85-88.
  • Baskonus, H.M., Bulut, H., and Atangana, A. 2016. On the complex hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Material and Structures, 25, 035022.
  • Biswas, A., Jawad, A.J.M. and Zhou, Q., 2018. Resonant optical solitons with anti-cubic nonlinearity. Optik, 157, 525-531.
  • Biswas, A. and Khalique, C.M., 2011. Stationary solutions for nonlinear dispersive Schrödinger's equation. Nonlinear Dynamics, 63, 623-626.
  • Biswas, A. and Konar, S., 2006. Introduction to non-Kerr law optical solitons, 1, CRC Press, Boca Raton FL, 27-54.
  • Biswas, A. and Khalique, C.M., 2011. Stationary solutions for nonlinear dispersive Schrödinger's equation. Nonlinear Dynamics, 63, 623-626.
  • Fan, E., 2000. Extented tanh-function method and its applications to nonlinear equations. Physics Letters A, 277, 212-218.
  • He, J.H. and Abdou, M.A., 2007. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons and Fractals, 34, 1421-1429.
  • Inan, I.E., Ugurlu, Y. and Inc, M., 2015. New Applications of the (G′/G,1/G)-Expansion Method. Acta Physica Polonica A, 128(3), 245-251.
  • Islam, Md. S., Khan, K. and Akbar, M.A., 2015. An analytical method for finding exact solutions of modified Korteweg-de Vries equation. Results in Physics, 5, 131-135.
  • Ismael, H.F., Bulut, H., Baskonus, H.M., Gao, W. 2020. Newly modified method and its application to thr coupled Boussinesq equation in ocean engineering with its linear stability analysis. Communications in Theoritical Physics, 72 (11), 115002.
  • Jawad, A.J., Mirzazadeh, M., Zhou, Q. and Biswas, A., 2017. Optical solitons with anti-cubic nonlinearity using three integration schemes. Superlattices and Microstructures, 105, 1-10.
  • Kaabar, M.K.A., Kaplan, M. and Siri, Z., 2021. New Exact Soliton Solutions of the (3+1)-Dimensional Conformable Wazwaz-Benjamin-Bona-Mahony Equation via Two Novel Techniques. Journal of Function Spaces, 465990.
  • Kaplan, M., Hosseini, K., Samadani, F., Raza, N. 2018. Optical soliton solutions of the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term. Journal of Modern Optics, 65(12) 1431-1436.
  • Kumar, D. and Kaplan, M., 2018. New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques. Chinese Journal of Physics, 56 (5), 2173-2185.
  • Kumar, D., Hosseini, and K., Samadani, F., 2017. The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, 439-446.
  • Lü, X., Tian, B., Zhang, H-Q., Xu, T. and Li, H., 2012. Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms. Nonlinear Dynamics, 67, 2279-2290.
  • Ma, W.X., Abdeljabbar, A. and Asaad, M.G., 2011. Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Applied Mathematics and Computation, 217, 10016-10023.
  • Ma, W.X.A and Lee, J.H., 2009. A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation. Chaos, Solitons and Fractals, 42, 1356-1363.
  • Mirzazadeh, M., Arnous, A.H., Mahmood, M.F., Zerrad, E. and Biswas, A., 2015. Soliton solutions to resonant nonlinear Schrödinger's equation with time-dependent coefficients by trial solution approach. Nonlinear Dynamics, 81, 277-282.
  • Roshid, H.O., Kabir, R.C., Bhowmik, R.C. and Datta, B.K., 2014. Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and exp(-Φ(ξ)) method. SpringerPlus, 3, 692.
  • Wang, M.L., 1995. Solitary wave solutions for variant Boussinesq equations. Physics Letters A, 199, 169-172.
  • Wazwaz, A.M. 2007. Multiple-soliton solutions for the Boussinesq equation. Applied Mathematics and Computation, 192 (2), 479-486.
  • Wazwaz, A.M.2004. The tanh method for travelling wave solutions of nonlinear equations. Applied Mathematics and Computation, 154 (3) ,713-723.
  • Yel, G. Baskonus, H.M. 2019. Solitons in conformable time-fractional Wu-Zhang system arising in coastal design. Pramana 92, 57.
  • Zayed, E.M.E., Alngar, M.E.M. and Al-Nowehy, A.G., 2019. On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms. Optik, 178 488-508.
There are 31 citations in total.

Details

Primary Language Turkish
Subjects Applied Mathematics
Journal Section Articles
Authors

Melike Kaplan 0000-0001-5700-9127

Publication Date February 28, 2022
Submission Date November 13, 2021
Published in Issue Year 2022 Volume: 22 Issue: 1

Cite

APA Kaplan, M. (2022). Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 22(1), 85-91. https://doi.org/10.35414/akufemubid.1023095
AMA Kaplan M. Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. February 2022;22(1):85-91. doi:10.35414/akufemubid.1023095
Chicago Kaplan, Melike. “Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22, no. 1 (February 2022): 85-91. https://doi.org/10.35414/akufemubid.1023095.
EndNote Kaplan M (February 1, 2022) Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22 1 85–91.
IEEE M. Kaplan, “Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 22, no. 1, pp. 85–91, 2022, doi: 10.35414/akufemubid.1023095.
ISNAD Kaplan, Melike. “Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22/1 (February 2022), 85-91. https://doi.org/10.35414/akufemubid.1023095.
JAMA Kaplan M. Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2022;22:85–91.
MLA Kaplan, Melike. “Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 22, no. 1, 2022, pp. 85-91, doi:10.35414/akufemubid.1023095.
Vancouver Kaplan M. Lineer Olmayan Kübik-Kuintik Schrödinger Denkleminin Üstel -Φ(ξ) Yöntemiyle Tam Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2022;22(1):85-91.