Fourth-order pattern forming PDEs: partial and approximate symmetries
Abstract
This paper considers pattern forming nonlinear models arising in the study of thermal convection and continuous media. A primary method for the derivation of symmetries and conservation laws is Noether’s theorem. However, in the absence of a Lagrangian for the equations investigated, we propose the use of partial Lagrangians within the framework of calculating conservation laws. Additionally, a nonlinear Kuramoto-Sivashinsky equation is recast into an equation possessing a perturbation term. To achieve this, the knowledge of approximate transformations on the admissible coefficient parameters is required. A perturbation parameter is suitably chosen to allow for the construction of nontrivial approximate symmetries. It is demonstrated that this selection provides approximate solutions.
Keyword : pattern formation, optimal system of one-dimensional subalgebras, Lie symmetries, exact solutions
How to Cite
Jamal, S., & Johnpillai, A. G. (2020). Fourth-order pattern forming PDEs: partial and approximate symmetries. Mathematical Modelling and Analysis, 25(2), 198-207. https://doi.org/10.3846/mma.2020.10115
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Patera and P. Winternitz. Subalgebras of real three- and four-dimensional Lie algebras. Math. Phys., 88:1449–1455, 1977. https://doi.org/10.1063/1.523441
Y . Pomeau and P. Manneville. Wave length selection in cellular flows. Phys. Lett., 75(A):296–298, 1980. https://doi.org/10.1016/0375-9601(80)90568-X
W. Sarlet. A comment on ’Conservation laws of higher order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives’. J. Phys. A Math. Theor., 43(45):458001, 2010. https://doi.org/10.1088/17518113/43/45/458001
G.I. Sivashinsky. Instabilities, pattern-formation, and turbulence in flames. Ann. Rev. Fluid Mech., 15:179–199, 1983. https://doi.org/10.1146/annurev.fl.15.010183.001143
J. Swift and P. Hohenberg. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15:319–328, 1977. https://doi.org/10.1103/PhysRevA.15.319
J. Belmonte-Beitia, V.M. P´erez-Garc´ıa, V. Vekslerchik and P.J. Torres. Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. Phys. Rev. Lett., 98(064102), 2007. https://doi.org/10.1103/PhysRevLett.98.064102
D.J. Benney. Long waves on liquid films. J. Math. and Phys., 45:150–155, 1966. https://doi.org/10.1002/sapm1966451150
G. Caginal and P.C. Fife. Higher order phase field models and detailed anisotropy. Phys. Rev. B, 34:4940–4943, 1986. https://doi.org/10.1103/PhysRevB.34.4940
P. Collet and J.P. Eckmann. Instabilities and fronts in extended systems. Princeton Series in Physics, Princeton University Press, New Jersey, 1990. https://doi.org/10.1515/9781400861026
G.T. Dee and W. van Saarloos. Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett., 60:2641–2644, 1988. https://doi.org/10.1103/PhysRevLett.60.2641
P. Holmes, J.L. Lumley and G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Univ. Press, Cambridge, 1996. https://doi.org/10.1017/CBO9780511622700
I. Hussain, F.M. Mahomed and A. Qadir. Approximate partial Noether operators of the Schwarzschild spacetime. J. Non. Math. Phys., 17(1):13–25, 2013. https://doi.org/10.1142/S1402925110000556
N.H. Ibragimov and V.F. Kovalev. Approximate and Renormgroup Symmetries. Springer-Verlag, Berlin, 2009. https://doi.org/10.1007/978-3-642-00228-1
S. Jamal. Solutions of quasi-geostrophic turbulence in multilayered configurations. Quaest. Math., 41:409–421, 2018. https://doi.org/10.2989/16073606.2017.1383947
S. Jamal and A. Mathebula. Generalized symmetries and recursive operators of some diffusive equations. Bull. Malays. Math. Sci. Soc., 42:697–706, 2019. https://doi.org/10.1007/s40840-017-0510-z
S.U. Jing-Rui, Z. Shun-Li and L. Ji-Na. Approximate Noether-type symmetries and conservation laws via partial Lagrangians for nonlinear wave equation with damping. Comm. Theor. Phys., 53:37–42, 2010. https://doi.org/10.1088/02536102/53/1/08
A.G. Johnpillai, K.S. Mahomed, C. Harley and F.M. Mahomed. Noether symmetry analysis of the dynamic Euler-Bernoulli beam equation. Z. Naturforsch, 71(5):447–456, 2016. https://doi.org/10.1515/zna-2015-0292
A.H. Kara and F.M. Mahomed. Noether-type symmetries and conservation laws via partial Lagrangians. Non. Dyn., 45:367–383, 2006. https://doi.org/10.1007/s11071-005-9013-9
T. Kawahara and S. Toh. Pulse interactions in an unstable dissipativedispersive nonlinear system. Phys. Fluids, 31:2103–2111, 1987. https://doi.org/10.1063/1.866610
A.H. Khater and R.S. Temsah. Numerical solutions of the generalized KuramotoSivashinsky equation by Chebyshev spectral collocation methods. Comp. Math. Appl., 56:1465–1472, 2008. https://doi.org/10.1016/j.camwa.2008.03.013
Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys., 55:356, 1976. https://doi.org/10.1143/PTP.55.356
J. Lega, J.V. Moloney and A.C. Newell. Swift-Hohenberg equation for lasers. Phys. Rev. Lett., 73:2978–2981, 1994. https://doi.org/10.1103/PhysRevLett.73.2978
R. Naz. The applications of the partial Hamiltonian approach to mechanics and other areas. Int. J. Non. Mech., 86:1–6, 2016. https://doi.org/10.1016/j.ijnonlinmec.2016.07.009
R. Naz, F.M. Mahomed and A. Chaudhry. A partial Hamiltonian approach for current value Hamiltonian systems. Comm. Non. Sci. Num. Sim., 19(10):3600– 3610, 2014. https://doi.org/10.1016/j.cnsns.2014.03.023
M.C Nucci and G. Sanchini. Noether symmetries quantization and superintegrability of biological models. Symmetry, 8:1–9, 2016. https://doi.org/10.3390/sym8120155
P. Olver. Application of Lie Groups to Differential Equations. Springer, New York, 1993. https://doi.org/10.1007/978-1-4612-4350-2
S. Opanasenko, A. Bihlo and R.O. Popovych. Group analysis of general Burgers-Korteweg-de Vries equations. J. Math. Phys., 58:081511, 2017. https://doi.org/10.1063/1.4997574
J. Patera and P. Winternitz. Subalgebras of real three- and four-dimensional Lie algebras. Math. Phys., 88:1449–1455, 1977. https://doi.org/10.1063/1.523441
Y . Pomeau and P. Manneville. Wave length selection in cellular flows. Phys. Lett., 75(A):296–298, 1980. https://doi.org/10.1016/0375-9601(80)90568-X
W. Sarlet. A comment on ’Conservation laws of higher order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives’. J. Phys. A Math. Theor., 43(45):458001, 2010. https://doi.org/10.1088/17518113/43/45/458001
G.I. Sivashinsky. Instabilities, pattern-formation, and turbulence in flames. Ann. Rev. Fluid Mech., 15:179–199, 1983. https://doi.org/10.1146/annurev.fl.15.010183.001143
J. Swift and P. Hohenberg. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15:319–328, 1977. https://doi.org/10.1103/PhysRevA.15.319