Symmetry properties of conservation laws for nonlinear Fokker-Planck equation describing cell population growth

Abstract

Symmetry properties of conservation laws of the (2+1)-dimensional nonlinear Fokker-Planck equation describing the growing cell populations are studied. Firstly, a complete symmetry classification is performed with respect to an equivalence transformation. A two-dimensional optimal system of Lie subalgebras is constructed and used to transform the equations into ordinary differential equations. Secondly, all nontrivial conservation laws are determined via multiplier method and their symmetry properties including symmetry homogeneity and symmetry invariance are investigated in detail. Finally, the Fokker-Planck equation with exponential diffusion is further studied where some invariant solutions and all nontrivial conservation laws are found. In particular, a power series solution is constructed and shown to converge to certain invariant solutions by choosing proper parameters. Moreover, the symmetry invariant conservation laws are transformed into the conservation laws or first integrals of the reduced equations. Consequently, double reductions are performed and some exact solutions are found.