Symmetry Reduction and Exact Solutions of A (3+1)- Dimensional Kadomtsev-Petviashvi-li-Benjamin-Bona-Mahony Equation

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2022 by IJMTT Journal
Volume-68 Issue-6
Year of Publication : 2022
Authors : Shuai Zhou
 10.14445/22315373/IJMTT-V68I6P523

How to Cite?

Shuai Zhou, " Symmetry Reduction and Exact Solutions of A (3+1)- Dimensional Kadomtsev-Petviashvi-li-Benjamin-Bona-Mahony Equation ," International Journal of Mathematics Trends and Technology, vol. 68, no. 6, pp. 180-189, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I6P523

Abstract
By applying a direct symmetry method, we get the symmetry group of the (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KPBBM) equation. Using the associated vector fields of the obtained symmetry, we get the optimal system of group-invariant solutions. To every case of the optimal system, we derive the reductions and some exact solutions of the (3+1)-dimensional KPBBM equation.

Keywords : Direct symmetry method, (3+1)-dimensional KPBBM equation, Optimal system, Exact solutions.

Reference

[1] S. Xu, J. He and Wang L, “The Darboux Transformation of the Derivative Nonlinear Schrodinger Equation,” Journal of Physics A Mathematical and Theoretical, vol. 44, no. 30, pp. 6629-6636, 2011.
[2] A. Barari, A.R. Ghotbi, and F. Farrokhzad, “Variational Iteration Method and Homotopy-Perturbation Method for Solving Different Types of Wave Equations,” Journal of Applied Sciences, vol. 8, no. 1, pp. 120-126, 2008.
[3] H. Alatas, A.A. Kandi, and A.A. Iskandar, “New Class of Bright Spatial Solitons Obtained by Hirota’s Method from Generalized Coupled Mode Equations of Nonlinear Optical Bragg Grating,” Journal of Nonlinear Optical Physics and Materials, vol. 17, no. 2, pp. 225-233, 2008.
[4] X.W. Chen, Y.M. Li, and Y.H. Zhao, “Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of Lagrange System,” Physics Letters A, vol. 37, no. 4-6, pp. 274-278, 2005.
[5] T, Mookum, and M, Khebchareon, “Finite Difierence Methods for Finding a Control Parameter in Two-Dimensional Parabolic Equation with Neumann Boundary Conditions,” Thai Journal of Mathematics, vol. 6, no. 1, pp. 117-137, 2008.
[6] H, Zhang. G.M. Wei, and Y.T. Gao, “On the General Form of the Benjamin-Bona-Mahony Equation in Fluid Mechanics,” The Journal of Chemical Physics, vol. 52, pp. 373-377, 2002.
[7] S. Ming, C. Yang, and B, Zhang, “Exact Solitary Wave Solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony Equation,” Applied Mathe- Matics and Computation, vol. 217, no. 4, pp. 1334-1339, 2010.
[8] Y. Yin, B. Tian, and X.Y. Wu, “Lump Waves and Breather Waves for a (3+1)-Dimensional Generalized Kadomtsev-PetviashviliBenjamin-Bona-Mahony Equation for an Offshore Structure,” Modern Physics Letters B, vol. 32, no. 10, pp. 1850031, 2018.
[9] A. Mekki, and M.M, Ali, Numerical Simulation of Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equations Using Finite Difference Method,” App- lied Mathematics and Computation, 2013.
[10] K.H. Tariq, A.R, “Soliton Solution of (3+1)-Dimensional Korteweg-De Vries Benjiamin-Mahony, Kadomtsov-Petviashvili-BenjaminBona-Mahony and Modified Korteweg-de Vries-Zakharov-Kuznetsov Equations and Their Applications in Water Waves,” Journal of King Saud University Science, vol. 31, no. 1, pp. 8-13, 2019.
[11] K.H. Tariq, and A.R. Seadawy, “Soliton Solutions of (3+1)-Dimensional Korteweg-de VriesBenjiamin-Bona-Mahony, KadomtsevPetviashvili-Benj- amin-Bona-Mahony and Modified Korteweg-de Vries-Zakharov-Kuznetsov Equations and Their Applications in Water Waves,” Journal of King Saud University Science, vol. 31, no. 1, pp. 8-13, 2019.
[12] C. Hu, R. Wang, and D.H. Ding, “Symmetry Groups, Physical Property Tensors, Elasticity and Dislocations in Quasicrystals,” Reports on Progress in Physics, vol. 63, no. 1, pp. 1, 2000.
[13] H.C. Ma, and S.Y. Lou, “Non-Lie Symmetry Groups of (2+1)-Dimensional Nonlinear Systems,” Communications in Theoretical Physics, vol. 46, no. 12, pp. 1005-1010, 2006.
[14] Z.Z. Dong, Y. Chen, and L. Wang, “Similarity Reductions of (2+1)-Dimensional Multicomponent Broer Kaup System,” Communications in Theoretical Physics, vol. 50, pp. 803-808, 2008.
[15] X.P. Xin, X.Q. Liu, and L.L. Zhang, “Symmetry Reductions and Exact Solutions of a (2+1)-Dimensional Nonlinear Evolution Equation,” Journal of Jinggangshan University, vol. 29, no. 4, pp. 411-416, 2012.
[16] H.C. Hu, J.B. Wang, and H.D. Zhu, “Symmetry Reduction of (2+1)-Dimensional Lax Kadomtsev-Petviashvili Equation,” Communications in Theoretical Physics, 2015.
[17] K. Sachin, and R. Setu, “Study of Exact Analytical Solutions and Various Wave Profiles of a New Extended (2+1)-Dimensional Boussinesq Equation Using Symmetry Analysis,” Journal of Ocean Engineering and Science, 2021.
[18] K. Ssachin, A. Hassan, and K.D. Shubham, “A Study of Bogoyavlenskiis (2+1)-Dimensional Breaking Soliton Equation: Lie Symmetry, Dynamical Behaviors and Closed-form Solutions,” Results in Physics, vol. 29, pp. 2211-7797, 2021.
[19] J. Zhuang, Y Liu, and P. Zhuang, “Variety Interaction Solutions Comprising Lump Solitons for the (2+1)-Dimensional Caudrey-DoddGibbon-Kotera-Sawada Equation”, AIMS Mathematics, vol. 6, no. 5, pp. 5370-5386, 2021.
[20] C.J. Bai, and H. Zhao, “A New Rational Approach to Find Exact Analytical Solutions to a (2+1)-Dimensional Symmetry,” Communications in Theoretical Physics, vol. 48, pp. 801-810, 2007.
[21] S.Y. Wang, F.X. Mei, “Form Invariance and Lie Symmetry of Equations of Non-Holonomic Systems,” Chinese Physics, vol. 11, no. 1, pp. 5, 2002.
[22] R. K. Gazizov, N.H. Ibragimov, “Lie Symmetry Analysis of Differential Equations in Finance,” Nonlinear Dynamics, vol. 17, no. 4, pp. 387-407, 1998.
[23] X.M. Feng, “Lie Symmetry and the Conserved Quantity of a Generalized Hamiltonian System,” Acta Physica Sinica, vol. 52, no. 5, pp. 1048-1050, 2003.
[24] S.K. Luo, “Mei Symmetry, Noether Symmetry and Lie Symmetry of Hamiltonian System,” Acta Physica Sinica, 2003.
[25] M. Craddock, K.A. Lennox, “The Calculation of Expectations for Classes of Diffusion Processes by Lie Symmetry Methods,” The Annals of Applied, 2021