Generalised two-component modified weakly dissipative Dullin-Gottwald-Holm system: invariance analysis and conservation laws
Abstract
The Dullin-Gottwald-Holm equation models the unidirectional propagation of shallow regime water waves. In this work, the Lie symmetry analysis of the generalised two-component modified weakly dissipative Dullin-Gottwald-Holm system is performed. Using symmetry reduction, the exact solutions are obtained in the form of power series and trigonometric functions. Also using multiplier approach, the conservation laws are obtained. The 3D graphical representations are also shown for obtained solutions.
Keyword : weakly dissipative Dullin-Gottwald-Holm system, Lie symmetries, exact solutions, conservation laws
How to Cite
Kumar, S., & Jyoti, D. (2022). Generalised two-component modified weakly dissipative Dullin-Gottwald-Holm system: invariance analysis and conservation laws. Mathematical Modelling and Analysis, 27(1), 101–116. https://doi.org/10.3846/mma.2022.14249
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L. Wei and Y. Wang. Symmetry analysis, conserved quantities and applications to a dissipative DGH equation. Journal of Differential Equations, 266(6):3189– 3208, 2019. https://doi.org/10.1016/j.jde.2018.08.055
P. Zhai, Z. Guo and W. Wang. Wave breaking phenomenon for a modified twocomponent Dullin-Gottwald-Holm equation. J. Math. Phys., 55(9):093101, 2014. https://doi.org/10.1063/1.4894368
R. Camassa and D.D. Holm. An integrable shallow water equation with peaked solitons. Physical review letters, 71(11):1661–1664, 1993. https://doi.org/10.1103/PhysRevLett.71.1661
W. Cheng and T. Xu. Blow-up of solutions to a modified two-component Dullin-Gottwald-Holm system. Applied Mathematics Letters, 105:106307, 2020. https://doi.org/10.1016/j.aml.2020.106307
W. Cheng and T. Xu. Local-in-space blow-up and symmetry of traveling wave solutions to a generalized two-component Dullin-Gottwald-Holm system. Monatshefte fu¨r Mathematik, 193(3):573–589, 2020. https://doi.org/10.1007/s00605-020-01411-w
P.A. Clarkson, E.L. Mansfield and T.J. Priestley. Symmetries of a class of nonlinear third-order partial differential equations. Mathematical and Computer Modelling, 25(8-9):195–212, 1997. https://doi.org/10.1016/S0895-7177(97)00069-1
A. Constantin and R.I. Ivanov. On an integrable two-component CamassaHolm shallow water system. Physics Letters A, 372(48):7129–7132, 2008. https://doi.org/10.1016/j.physleta.2008.10.050
H.R. Dullin, G.A. Gottwald and D.D. Holm. An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett., 87(19):194501, 2001. https://doi.org/10.1103/PhysRevLett.87.194501
F. Guo, H. Gao and Y. Liu. On the wave-breaking phenomena for the twocomponent Dullin–Gottwald–Holm system. J. London Math. Soc., 86(3):810– 834, 2012. https://doi.org/10.1112/jlms/jds035
R.K. Gupta and Anupma. The Dullin-Gottwald-Holm equation: Classical Lie approach and exact solutions. Int. J. Nonlinear Sci., 10(2):146–152, 2010. ISSN 1749-3889.
N.H. Ibragimov. A new conservation theorem. Journal of Mathematical Analysis and Applications, 333(1):311–328, 2007. https://doi.org/10.1016/j.jmaa.2006.10.078
D. Jyoti and S. Kumar. Exact non-static solutions of Einstein vacuum field equations. Chinese Journal of Physics, 68:735–744, 2020. https://doi.org/10.1016/j.cjph.2020.10.006
D. Jyoti and S. Kumar. Modified Vakhnenko–Parkes equation with power law nonlinearity: Painlev´e analysis, analytic solutions and conservation laws. The European Physical Journal Plus, 135:762, 2020. https://doi.org/10.1140/epjp/s13360-020-00785-y
D. Jyoti and S. Kumar. Invariant solutions and conservation laws of Einstein field equations in non-comoving radiation fields. Chinese Journal of Physics, 70:37–43, 2021. https://doi.org/10.1016/j.cjph.2020.12.018
D. Jyoti, S. Kumar and R.K. Gupta. Exact solutions of Einstein field equations in perfect fluid distribution using Lie symmetry method. The European Physical Journal Plus, 135:604,2020. https://doi.org/10.1140/epjp/s13360-020-00622-2
S. Kumar. Invariant solutions of Biswas-Milovic equation. Nonlinear Dyn., 87(2):1153–1157, 2016. https://doi.org/10.1007/s11071-016-3105-6
H. Liu and Y. Geng. Symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid. J. Differ. Equations, 254(5):2289–2303, 2013. https://doi.org/10.1016/j.jde.2012.12.004
R. Naz. Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J. Appl. Math., 2012:871253, 2012. https://doi.org/10.1155/2012/871253
R. Naz, I. Naeem and M. Khan. Conservation laws of some physical models via symbolic package GeM. Math. Probl. Eng., 2013:897912, 2013. https://doi.org/10.1155/2013/897912
P.J. Olver. Applications of Lie groups to differential equations. Springer, New York, 1986. https://doi.org/10.1007/978-1-4684-0274-2
W. Rudin. Principles of mathematical analysis, volume 3. McGraw-hill, New York, 1964.
S.F. Tian. Asymptotic behavior of a weakly dissipative modified twocomponent Dullin–Gottwald–Holm system. Appl. Math. Lett., 83:65–72, 2018. https://doi.org/10.1016/j.aml.2018.03.019
S.F. Tian. Infinite propagation speed of a weakly dissipative modified twocomponent Dullin–Gottwald–Holm system. Appl. Math. Lett., 89:1–7, 2019. https://doi.org/10.1016/j.aml.2018.09.010
S.F. Tian, J.J. Yang, Z.Q. Li and Y.R. Chen. Blow-up phenomena of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett., 106:106378, 2020. https://doi.org/10.1016/j.aml.2020.106378
L. Wei and Y. Wang. Symmetry analysis, conserved quantities and applications to a dissipative DGH equation. Journal of Differential Equations, 266(6):3189– 3208, 2019. https://doi.org/10.1016/j.jde.2018.08.055
P. Zhai, Z. Guo and W. Wang. Wave breaking phenomenon for a modified twocomponent Dullin-Gottwald-Holm equation. J. Math. Phys., 55(9):093101, 2014. https://doi.org/10.1063/1.4894368