Optimal Systems and Group Classification of (1+2)-Dimensional Heat Equation

The optimal systems and symmetry breaking interactions for the (1+2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H₂. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H₂. A list of representatives of all Lie subalgebras of the Lie algebra h₂ of the Lie group H₂ is given in the form of tables and many of their properties are established. We derive the most general interactions F(t,x,y,u,u_x,u_y) such that the equation u_t=u_xx+u_yy+F(t,x,y,u,u_x,u_y) is invariant under each subgroup.