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The transport of chemicals through soils to the groundwater or precipitation at the soils surfaces leads to degradation of these resources. Serious consequences may be suffered in the long run. In this paper, we consider macroscopic deterministic models describing contaminant transport in saturated soils under uniform radial water flow backgrounds. The arising convection-dispersion equation given in terms of the stream functions is analyzed using classical Lie point symmetries. A number of exotic Lie point symmetries are admitted. Group invariant solutions are classified according to the elements of the one-dimensional optimal systems. We analyzed the group invariant solutions which satisfy the physical boundary conditions.

In this paper, we consider a form of convection-dispersion equation given in terms of the stream functions. The governing equation describing movements of contaminants under radial water flow background is given in the conserved form. As such, the conserved form of the governing equation may be written as a system of first order partial differential equations referred to as an auxiliary system, by the introduction of the nonlocal variable (or the potential variable). The resulting system of equations admits a number of (local) point symmetries which induce the nonlocal symmetries for the original governing equation. We construct classes of exact solutions using admitted genuine nonlocal symmetries, which include the invariant solutions obtained via corresponding point symmetries of the governing equation. (2010): 35A08, 35B06, 35A20. Mathematics Subject Classification