Conservation laws and invariant solutions for soil water equations (Q1422550)

The reviewed article is devoted to Lie group analysis of the nonlinear partial differential equation \[ C(\psi)\psi_t =(K(\psi)\psi_x)_x+(K(\psi)(\psi_z-1))_z-S(\psi), \] for the simulation of soil water infiltration, redistribution and extraction in a bedded soil profile overlaying a shallow water table and irrigated by a line source drip irrigation system. Here \(\psi\) is a soil moisture pressure head, \(C(\psi)\) is the specific water capacity, \(K(\psi)\) is the unsaturated hydraulic conductivity, \(S(\psi)\) is a sink or source term. The authors determine associate symmetries with conservation laws and obtain exact solutions of this equation which are invariant under a three-dimensional subalgebra of the symmetry Lie algebra of the equation. The results of the article of \textit{V. A. Baikov, R. K. Gazizov, N. H. Ibragimov} and \textit{V. P. Kovalev} [Nonlinear Dyn. 13, No. 4, 395--409 (1997; Zbl 0887.76075)] are used.