Some exact solutions of (1+1)-dimensional Kaup-system and seventh-order Kawahara equation

Astha Chauhan; Rajan Arora

doi:10.26637/MJM0801/0025

Abstract

In this paper, we solve a $(1+1)$-dimensional Kaup system and seventh order Kawahara equation. The Lie symmetry analysis is used to perform the similarity reduction and to obtain the exact solutions of the $(1+1)$ dimensional Kaup system and seventh order Kawahara equation. Similarity transformation method reduces $(1+1)$-dimensional Kaup system into a system of ODEs and nonlinear Kawahara equation into nonlinear ordinary differential equation (ODE) and helps to find exact solutions. With the help of reduction equations, we have obtained the exact explicit solutions. Moreover, later by power series method, the exact analytic solutions for seventh order Kawahara equation are obtained.

Keywords:

Kaup system, Kawahara equation, Similarity transformation method, Infinitesimal generator, Similarity solutions

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Astha Chauhan Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, India.
Rajan Arora Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, India.

Pages: 151-158
Date Published: 01-01-2020
Vol. 8 No. 01 (2020): Malaya Journal of Matematik (MJM)

C. Rogers and W. K. Schief, Backlund and Darboux Transformation Geometry and Modern Applications in Solitons Theory, Camb. Uni., (2002).

A. M. Wazwaz, The tanh method: Solitons and Peroidic Solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough Equations, Chaos Solitons Fract., 35(2005), 55-63.

M. L. Wang, Solitary Wave Solutions For Variant Boussinesq Equations, Phys. Lett. A, 199(1995) 169-172.

G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations, Springer Verlag, New York, (1974), 143-166.

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York 107(1993).

P. Clarkson and M. Kruskal, New Similarity Reductions of the Boussinesq Equation, J. Math. Phys., 30(10)(1989), 2201-2213.

R. Hirota, Exact Solution of The Korteweg-de Vries Equation For Multiple Collisions of Solitons, Phys. Rev. Lett., 27(1971), 1192-1194.

C. K. Chen and S. H. Ho, Solving Partial Differential Equations by Two-Dimensional Differential Transform Method, Appl. Math. Comput., 106 (1999), 171-179.

S. Lie, Theorie der transformationsgruppeni, Math. Ann., 16(4):441, (1880).

R. Arora, Similarity Solutions and Evolution of Weak Discontinuities in a Vander Waals Gas, Can. Appl. Math. Q., 13(2005), 297-311.

R. Arora and V. D. Sharma, Convergence of Strong Shock in a Vander Waals Gas, SIAM J. Appl. Math., 66(2006), $1825-1837$.

T. Y. Hui, C. H. Lin and L. X. Qiang, Reduction and New Explicit Solutions of (2+1)-Dimensional Breaking Soliton Equation, Commun. Theor. Phys., 45(2006), 3335.

M. Kumar, R. Kumar and A, Kumar, On Similarity Solutions of Zabolotskava-Khokhlov Equation, Comput. Math. Appl., 68(2014), 454-463.

J. Q. Yu, X. Q. Liu and T. T. Wang, Exact Solutions and Conservation Laws of (2+1)-Dimensional Boiti-LeonPempinelli-Equation, Appl. Math. Comput., 216(2010), 2293-2300.[15] Y. H. Zhi, New Similarity Reduction Solutions for the(2+1)-Dimensional Nizhnik-Novikov-Veselov Equation, Commun. Theor. Phys., 59(2013), 263-267.

J. L. Bona, M. Chen and J. C. Saut, Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory, Nonlinear Sci., 12(2002), 283-318.

A. M. Wazwaz, Soliton Solutions for Seventh-Order Kawahara Equation With Time-Dependent Coefficients, Mod. Phys. Lett. B., 25(9)(2011), 643-648.

S. Ynag and C. Hua, Lie Symmetry Reductions and Exact Solutions of a Coupled KdV-Burgers Equation, Appl. Math. Comput., 234(2014), 579-583.

H. YANG, W. Liu, B. Yang and B. He, Lie Symmetry Analysis and Exact Explicit Solutions of Three Dimensional Kudryashov-Sinelshchikov Equation, Commun. Nonlinaer Sci. Numer. Simula., 27(2015), 271-280.

S. Sahoo, G. Garai and S. S. Ray, Lie Symmetry Analysis for Similarity Reduction and Exact Solutions of Modified KdV-Zakharov-Kuznetsov Equations, Nonlinear Dyn., 87(2017), 1995-2000.

H. Liu and J. Li, Lie Symmetry Analysis and Exact Solutions for the Short Pulse Equation, Nonlinear Anal., $71(2009)$, 2126-2123.

H. Liu, J. Li and L. Liu, Lie Symmetry Analysis, Optimal Systems and Exact Solutions to the Fifth-Order KdV Types of Equations, J. Math. Anal. Appl., 368(2010), 551558.

${ }^{[23]}$ H. Liu, J. Li and Q. Zhang, Lie Symmetry Analysis and Exact Explicit Solutions for General Burgers Equation, $J$. Comput. Appl. Math., 228(2009), 1-9.

Y. Zhang, Lie Symmetry Analysis and Exact Solutions of the Sharma-Tasso-Olever Equation, Int. J. Appl. Math., $46(2)(2016), 17-24$.

G. Wang, A. H. Kara, K. Fakhar, J. V. Guzman and A. Bisas, Group Analysis, Exact Solutions and Conservation Laws of a Generalized Fifth Order KdV Equation, Chaos SolitpnsFract., 86(2016), 8-15.

B. Gao and H. Tian, Symmetry Reductions and Exact Solutions to the Ill-Posed Boussinesq Equation, Int. J. Nonlinear Mech., 72(2015), 80-83.