Exact solutions to a nonlinear reaction-diffusion equation and hyperelliptic integrals inversion

An approach is proposed to obtain some exact explicit stationary solutions in terms of elliptic functions to a nonlinear reaction-diffusion equation. The method is based on the reduction of the hyperelliptic integral to the elliptic one and its inversion via the Weierstrass and Jacobi elliptic functions. The solutions for both polynomial reaction and diffusion functions include bounded periodic and localized (in space) functions. Such solutions seem to be the best candidates to describe periodic nanostructures observed in experiments on formation of thin films by means of molecular epitaxy (the so-called 'quantum wires'). Generalization of the approach is discussed for reaction and diffusion functions distinctive from polynomials. In particular, explicit stationary solutions are found in terms of elliptic functions for arbitrary diffusion and relevant reaction terms.