Abstract
The dynamics of localised solitary wave solutions play an essential role in the fields of mathematical sciences such as optical physics, plasma physics, nonlinear dynamics and many others. The prime objective of this study is to obtain localised solitary wave solutions and exact closed-form solutions of the \((2+1)\)-dimensional universal hierarchy equation (UHE) using the Lie symmetry approach. Besides, the Lie infinitesimals, all the vector fields, commutation relations of Lie algebra and symmetry reductions are derived via the Lie transformation method. Meanwhile, the universal hierarchy equation is reduced into nonlinear ODEs through two stages of symmetry reductions. The closed-form invariant solutions are attained under some parametric conditions imposed on infinitesimal generators. Because of the presence of arbitrary independent functional parameters and other constants, these group-invariant solutions are explicitly displayed in terms of arbitrary functions that are more relevant, beneficial and useful for explaining nonlinear complex physical phenomena. Furthermore, the dynamical structures of the obtained exact solutions are illustrated for suitable values of arbitrary constants through 3D-plots based on numerical simulation. Some of these localised solitary waves are double solitons, periodic lump solitons, dark solitons, five-solitons, hemispherical solitons and lump-type solitons.
Similar content being viewed by others
References
S Zhang, Chaos Solitons Fractals 31, 951 (2007)
N Taghizadeh and M Mirzazadeh, Appl. Appl. Math. 6(11), 1893 (2011)
S Zhang, Chaos Solitons Fractals 30, 1213 (2006)
P B Xu, Y T Gao, Y Xin and L G Dong, Commun. Theor. Phys. 55(6), 1017
A M Wazwaz, Math. Comput. Model. 45, 473 (2007)
M A Abdou, Nonlinear Dynam. 52, 1 (2008)
S Zhang, Appl. Math. Comput. 216, 1546 (2010)
S Zhang and T C Xia, Appl. Math. Comput. 183, 1190 (2006)
J Lin, S Y Lou and K L Wang, Chin. Phys. Lett. 18, 1173 (2001)
L Song and H Q Zhang, Appl. Math. Comput. 197, 87 (2008)
L Song and H Q Zhang, Appl. Math. Comput. 187, 1373 (2007)
H Y Zhi, Appl. Math. Comput. 210, 530 (2009)
P B Xu, Y T Gao, X L Gai, D X Meng, Y J Shen and L Wang, Appl. Math. Comput. 218, 2489 (2011)
H W Hu and J Yu, Chin. Phys. B 21(2), 020202 (2012)
A H Salas, Appl. Math. Comput. 217, 7391 (2011)
S Kumar, M Kumar and D Kumar, Pramana – J. Phys. 94: 28 020202(2020)
S Kumar and A Kumar, Nonlinear Dyn. 98(3), 1891 (2019)
S Kumar and D Kumar, Comput. Math. Appl. 77(8), 2096 (2019)
S Kumar and S Rani , Pramana – J. Phys. 94: 116 (2020)
S Kumar and A Kumar, Mod. Phys. Lett. B 34(01), 2150015 (2020)
S Kumar, A Kumar and H Kharbanda, Phys. Scr. 95(6), 065207 (2020)
S Kumar M Niwas and A M Wazwaz, Phys. Scr. 95(9), 095204 (2020)
S Kumar, L Kaur and M Niwas, Chin. J. Phys. 71, 518 (2021)
S Kumar, D Kumar and A Kumar, Chaos Solitons Fractals 142, 110507 (2021)
S Kumar, D Kumar and H Kharbanda, Pramana – J. Phys. 95: 33 (2021)
S Kumar and S Rani, Pramana – J. Phys. 95: 51 (2021)
H Baran, I S Krasischik, O I Morozov and P Vojcak, J. Nonlinear Math. Phys. 21(4), 643 (2014)
H Baran, I S Krasischik, O I Morozov and P Vojcak, J. Nonlinear Math. Phys. 22(2), 210 (2015)
A Lelito and O I Morozov, J. Geom. Phys. 131, 89 (2018)
H Baran, I S Krasischik, O I Morozov and P Vojcak, Theor. Math. Phys. 196, 1089 (2018)
P J Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1986)
X Hu, Y Li and Y Chen, J. Math. Phys. 56, 1 (2015)
G W Bluman and S Kumei, Symmetries (Springer Science & Business Media, 1989)
L Kaur and A M Wazwaz, Nonlinear Dynam. 94(4), 2469 (2018)
L Kaur and A M Wazwaz, Waves Random Complex Media 3(2), 199 (2019)
L Kaur and R K Gupta, Appl. Math. Comput. 231, 560 (2014)
L Kaur and R K Gupta, Math. Model Appl. Sci. 36(5), 584 (2013)
Acknowledgements
This work is funded by Science and Engineering Research Board, SERB-DST, India under project scheme Empowerment and Equity Opportunities for Excellence in Science (EEQ/2020/000238). The author, Sachin Kumar, has received this research grant.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kumar, S., Niwas, M. Exact closed-form solutions and dynamics of solitons for a \((2+1)\)-dimensional universal hierarchy equation via Lie approach. Pramana - J Phys 95, 195 (2021). https://doi.org/10.1007/s12043-021-02219-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-021-02219-5
Keywords
- Closed-form solutions
- solitons
- Lie symmetry approach
- \((2+1)\)-dimensional universal hierarchy equation