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

## TL;DR

This paper investigates how nonlinear boundary conditions influence advection-diffusion equations on the half-line, with applications to crystal growth, focusing on the role of symmetric solutions in the system's global behavior.

## Contribution

It introduces a model incorporating nonlinear boundary fluxes with gauge symmetry, analyzing the impact of symmetric solutions on the dynamics of crystal growth models.

## Key findings

- Discrete gauge symmetry affects boundary flux behavior.
- Periodic, homoclinic, and heteroclinic solutions influence global dynamics.
- Nonlinear boundary conditions model non-adiabatic effects in 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.