Advection-diffusion dynamics with nonlinear boundary flux as a model for crystal growth

Abstract

We analyze the effect of nonlinear boundary conditions on an advection-diffusion equation on the half-line. Our model is inspired by models for crystal growth where diffusion models diffusive relaxation of a displacement field, advection is induced by apical growth, and boundary conditions incorporate non-adiabatic effects on displacement at the boundary. The equation, in particular the boundary fluxes, possesses a discrete gauge symmetry, and we study the role of simple, entire solutions, here periodic, homoclinic, or heteroclinic relative to this gauge symmetry, in the global dynamics.