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Symmetry group classification for general Burgers equation. (English) Zbl 1222.35195
Summary: The present paper solves the problem of the group classification of the general Burgers’ equation ${u}_{t}=f\left(x,u\right){u}_{x}^{2}+g\left(x,u\right){u}_{xx}$, where $f$ and $g$ are arbitrary smooth functions of the variable $x$ and $u$, by using the Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification that is called preliminary group classification. Looking at the adjoint representation of ${G}_{ℰ}$ on its Lie algebra ${𝔤}_{5}$, we will deal with the construction of the optimal system of its one-dimensional subalgebras. The result of the work is a wide class of equations summarized in table form.
MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 35K55 Nonlinear parabolic equations 35A30 Geometric theory for PDE, characteristics, transformations
References:
 [1] Lie, S.: On integration of a class of linear partial differential equations by means of definite integrals, Arch math 6, 328 (1881) [2] Ovsiannikov, L. V.: Group analysis of differential equations, (1982) · Zbl 0485.58002 [3] Ibragimov, N. H.; Tottisi, M.; Valenti, A.: Preliminary group classification of equations $utt=f\left(x,ux\right)uxx+g\left(x,ux\right)$, J math phys 32, No. 11, 2988-2995 (1991) · Zbl 0737.35099 · doi:10.1063/1.529042 [4] Song, L.; Zhang, H.: Preliminary group classification for the nonlinear wave equation $utt=f\left(x,u\right)uxx+g\left(x,u\right)$, Nonlinear anal (2008) [5] Ibragimov, N. H.; Tottisi, M.; Valenti, A.: Differential invariants of nonlinear equations $utt=f\left(x,ux\right)uxx+g\left(x,ux\right)$, Commun nonlinear sci numer simul 9, 6980 (2004) · Zbl 1046.35074 · doi:10.1016/j.cnsns.2003.09.001 [6] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M.: Method for solving the kortewegde Vries equation, Phys rev lett 19, 10951097 (1967) · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095 [7] Li, Y. S.: Soliton and integrable systems, (1999) [8] Hirota, R.; Satsuma, J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the tota lattice, Suppl prog theor phys 59, 64100 (1976) [9] Olver, P. J.: Applications of Lie group to differential equations, Graduate text maths 107 (1986) · Zbl 0588.22001 [10] Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989) [11] Cantwell, B. J.: Introduction to symmetry analysis, (2002) [12] Liu, H.; Li, J.; Zhangb, Q.: Lie symmetry analysis and exact explicit solutions for general Burgers equation, J comput appl math (2008) [13] Gandarias, M. L.; Torrisi, M.; Tracina, R.: On some differential invariants for a family of diffusion equations, J phys A: math theor 40, 8803-8813 (2007) · Zbl 1121.35006 · doi:10.1088/1751-8113/40/30/013 [14] Torrisi, M.; Tracina, R.: Second-order differential invariants of a family of diffusion equations, J phys A: math gen 38, 7519-7526 (2005) · Zbl 1078.35051 · doi:10.1088/0305-4470/38/34/006 [15] Gandarias, M. L.; Torrisi, M.; Valenti, A.: Symmetry classification and optimal systems of a non-linear wave equation, Int J nonlinear mech 39, 389398 (2004) [16] Maluleke, G. H.; Mason, D. P.: Optimal system and group invariant solutions for a nonlinear wave equation, Commun nonlinear sci numer simul 9, 93104 (2004) · Zbl 1036.35010 · doi:10.1016/S1007-5704(03)00018-2 [17] Olver, P. J.: Equivalence, invariants, and symmetry, (1995) · Zbl 0837.58001