Similarity solutions of an equation describing ice sheet dynamics

This paper focuses upon the derivation of the similarity solutions of a nonlinear equation associated with a free boundary problem arising in glaciology. We present a potential symmetry analysis of this second order nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. A full classical and also a non-classical symmetry analysis are presented. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties of the free moving boundary in the presence of an accumulation-ablation function with realistic physical properties.