On the classification of similarity solutions of a two-dimensional diffusion-advection equation (Q883958)

The authors consider the two-dimensional diffusion-advection equation \(u_t=u_{yy}-k'(u)u_x\). They classify the possible symmetry algebras depending on \(k(u)\) and determine in each case the adjoint action in order to derive an optimal system and the similarity variables. The largest symmetry algebra is obtained for \(k(u)=u^2\). For this case all possible symmetry reductions are determined. For some reductions explicit solutions are given. Finally, some numerical similarity solutions are presented.