The group classification of models of axion electrodynamics with arbitrary self interaction of axionic field is carried out. It is shown that extensions of the basic Poincaré invariance of these models appear only for constant and exponential interactions. The related conservation laws are discussed. The maximal continuous symmetries of the 3d Chern-Simons electrodynamics and Carroll-Field-Jackiw electrodynamics are presented. Using the Inönü-Wigner contraction the nonrelativistic limit of equations of axion electrodynamics is found. Exact solutions for the electromagnetic and axion fields are discussed including those which describe propagation with group velocities faster than the speed of light. However these solutions are causal since the corresponding energy velocities are subluminal.
Using the three-dimensional subalgebras of the Lie algebra of Poincaré group an extended class of exact solutions for the field equations of the axion electrodynamics is obtained. These solutions include arbitrary parameters and arbitrary functions as well. The most general solutions include six arbitrary functions. Among them there are bound and square integrable solutions which propagate faster than light. However, their energy velocities are smaller than the velocity of light.
An exhaustive group classification of variable coefficient generalized Kawahara equations is carried out. As a result, we derive new variable coefficient nonlinear models admitting Lie symmetry extensions. All inequivalent Lie reductions of these equations to ordinary differential equations are performed. We also present some examples on the construction of exact and numerical solutions.
We classify the Lie symmetries of variable coefficient Gardner equations (called also the combined KdV-mKdV equations). In contrast to the particular results presented in [M. Molati, M.P. Ramollo, Commun. Nonlinear Sci. Numer. Simulat. 15 (2012), 1542-1548] we perform the exhaustive group classification. It is shown that the complete results can be achieved using either the gauging of arbitrary elements of the class by the equivalence transformations or the method of mapping between classes. As by-product of the second approach the complete group classification of a class of variable coefficient mKdV equations with forcing term is derived. Advantages of the use of the generalized extended equivalence group in comparison with the usual one are also discussed.
We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg-de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.
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