Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media

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Abstract
The equation for the horizontal transport of a liquid in an unsaturated scale-heterogeneous porous medium is ∂θ/∂t=λ(x)∂/∂x[C(θ)∂θ/∂x] −λ′(x)E(θ)∂θ/∂x−λ″(x)∫(C+E)dθ. A systematic search for Lie–Bäcklund symmetries leads to the requirement that C=a(b−θ)−2, as in the homogeneous (λ=1) case. More generally, (λ,E) may be ((1+mx)α, (1/α− (3)/(2) )C) or (exp(mx),−3C/2). In these cases the transport equation may be linearized and solved exactly. Examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for Burgers’ equation.