Home Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation
Article
Licensed
Unlicensed Requires Authentication

Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation

  • Jicheng Yu ORCID logo EMAIL logo and Yuqiang Feng ORCID logo
Published/Copyright: October 2, 2024

Abstract

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

MSC 2020: 76M60; 34K37

References

[1] A. Bansal, A. Biswas, Q. Zhou and M. M. Babatin, Lie symmetry analysis for cubic-quartic nonlinear Schrödinger’s equation, Optik 169 (2018), 12–15. 10.1016/j.ijleo.2018.05.030Search in Google Scholar

[2] Y. Feng and J. Yu, Lie symmetry analysis of fractional ordinary differential equation with neutral delay, AIMS Math. 6 (2021), no. 4, 3592–3605. 10.3934/math.2021214Search in Google Scholar

[3] Y. Q. Feng and J. C. Yu, Lie group method for constructing integrating factors of first-order ordinary differential equations, Int. J. Math. Ed. Sci. 54 (2023), no. 2, 292–308. 10.1080/0020739X.2021.1992027Search in Google Scholar

[4] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/9789812817747Search in Google Scholar

[5] Q. Hussain, F. D. Zaman and A. H. Kara, Invariant analysis and conservation laws of time fractional Schrödinger equations, Optik 206 (2020), Article ID 164356. 10.1016/j.ijleo.2020.164356Search in Google Scholar

[6] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 1, CRC Press, Boca Raton, 1993. Search in Google Scholar

[7] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007), no. 1, 311–328. 10.1016/j.jmaa.2006.10.078Search in Google Scholar

[8] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A 44 (2011), Article ID 432002. 10.1088/1751-8113/44/43/432002Search in Google Scholar

[9] A. Jhangeer, H. Rezazadeh and A. Seadawy, A study of travelling, periodic, quasiperiodic and chaotic structures of perturbed Fokas–Lenells model, Pramana-J. Phys. 95 (2021), Paper No. 41. 10.1007/s12043-020-02067-9Search in Google Scholar

[10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Search in Google Scholar

[11] A. M. Nass, Symmetry analysis of space-time fractional Poisson equation with a delay, Quaest. Math. 42 (2019), no. 9, 1221–1235. 10.2989/16073606.2018.1513095Search in Google Scholar

[12] P. J. Olver, Applications of Lie Groups to Differential Equations, Grad. Texts in Math. 107, Springer, New York, 1986. 10.1007/978-1-4684-0274-2Search in Google Scholar

[13] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[14] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993. Search in Google Scholar

[15] A. R. Seadawy, Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma, Comput. Math. Appl. 67 (2014), no. 1, 172–180. 10.1016/j.camwa.2013.11.001Search in Google Scholar

[16] A. R. Seadawy, K. K. Ali and R. I. Nuruddeen, A variety of soliton solutions for the fractional Wazwaz–Benjamin–Bona–Mahony equations, Results Phys. 12 (2019), 2234–2241. 10.1016/j.rinp.2019.02.064Search in Google Scholar

[17] A. R. Seadawy and B. Alsaedi, Contraction of variational principle and optical soliton solutions for two models of nonlinear Schrödinger equation with polynomial law nonlinearity, AIMS Math. 9 (2024), no. 3, 6336–6367. 10.3934/math.2024309Search in Google Scholar

[18] A. R. Seadawy and B. A. Alsaedi, Soliton solutions of nonlinear Schrödinger dynamical equation with exotic law nonlinearity by variational principle method, Opt. Quant. Electron. 56 (2024), Paper No. 700. 10.1007/s11082-024-06367-xSearch in Google Scholar

[19] A. R. Seadawy, M. Arshad and D. Lu, The weakly nonlinear wave propagation theory for the Kelvin–Helmholtz instability in magnetohydrodynamics flows, Chaos Solitons Fractals 139 (2020), Article ID 110141. 10.1016/j.chaos.2020.110141Search in Google Scholar

[20] A. R. Seadawy, S. T. R. Rizvi, I. Ali, M. Younis, K. Ali, M. M. Makhlouf and A. Althobaiti, Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen–Lee–Liu model, Opt. Quant. Electron. 53 (2021), Paper No. 172. 10.1007/s11082-021-02823-0Search in Google Scholar

[21] J. Wang, K. Shehzad, A. R. Seadawy, M. Arshad and F. Asmat, Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov-Kuznetsov systems with their stability, J. Taibah Univ. Sci. 17 (2023), no. 1, Article ID 2163872. 10.1080/16583655.2022.2163872Search in Google Scholar

[22] M. Yourdkhany and M. Nadjafikhah, Symmetries, similarity invariant solution, conservation laws and exact solutions of the time-fractional harmonic oscillator equation, J. Geom. Phys. 153 (2020), Article ID 103661. 10.1016/j.geomphys.2020.103661Search in Google Scholar

[23] J. Yu, Lie symmetry analysis of time fractional Burgers equation, Korteweg–de Vries equation and generalized reaction-diffusion equation with delays, Int. J. Geom. Methods Mod. Phys. 19 (2022), no. 14, Article ID 2250219. 10.1142/S021988782250219XSearch in Google Scholar

[24] J. Yu, Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion, J. Appl. Anal. 30 (2024), no. 1, 137–145. 10.1515/jaa-2023-0107Search in Google Scholar

[25] J. Yu and Y. Feng, Lie symmetry analysis and exact solutions of space-time fractional cubic Schrödinger equation, Int. J. Geom. Methods Mod. Phys. 19 (2022), no. 5, Article ID 2250077. 10.1142/S0219887822500773Search in Google Scholar

[26] J. Yu and Y. Feng, Lie symmetry, exact solutions and conservation laws of some fractional partial differential equations, J. Appl. Anal. Comput. 13 (2023), no. 4, 1872–1889. 10.11948/20220268Search in Google Scholar

[27] J. Yu and Y. Feng, Group classification for one type of space-time fractional quasilinear parabolic equation, Quantum Stud. Math. Found. 1 (2024), 10.1007/s40509-024-00338-3. 10.1007/s40509-024-00338-3Search in Google Scholar

[28] J. Yu and Y. Feng, Lie symmetry analysis, power series solutions and conservation laws of ( 2 + 1 ) -dimensional time fractional modified Bogoyavlenskii–Schiff equations, J. Nonlinear Math. Phys. 31 (2024), no. 1, Paper No. 27. 10.1007/s44198-024-00195-zSearch in Google Scholar

[29] J. Yu and Y. Feng, On the generalized time fractional reaction-diffusion equation: Lie symmetries, exact solutions and conservation laws, Chaos Solitons Fractals 182 (2024), Article ID 114855. 10.1016/j.chaos.2024.114855Search in Google Scholar

[30] J. Yu, Y. Feng and X. Wang, Lie symmetry analysis and exact solutions of time fractional Black–Scholes equation, Int. J. Financ. Eng. 9 (2022), no. 4, Article ID 2250023. 10.1142/S2424786322500232Search in Google Scholar

[31] Z.-Y. Zhang and G.-F. Li, Lie symmetry analysis and exact solutions of the time-fractional biological population model, Phys. A 540 (2020), Article ID 123134. 10.1016/j.physa.2019.123134Search in Google Scholar

Received: 2024-05-02
Revised: 2024-07-19
Accepted: 2024-08-23
Published Online: 2024-10-02

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 21.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2024-0072/html
Scroll to top button