Exact solutions of the nonlinear diffusion equation $u_t=\Delta\log u+ \lambda u$ in a two-dimensional coordinate space

Investigating exact solutions of the nonlinear diffusion equation, which arises in wave problems in hydrodynamics and heat transfer processes, we find new classes of solutions of this equation in a two-dimensional coordinate space and demonstrate a special superposition principle that allows constructing complex multimode solutions from the simplest two- and three-mode solutions.