Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations

TL;DR

This paper derives new stationary solutions for driven sixth- and fourth-order Cahn-Hilliard equations, revealing detailed asymptotic behavior and solution structures relevant to epitaxial nano-structure growth.

Contribution

It extends the analytical solution set of driven Cahn-Hilliard equations using matched asymptotic expansions and phase space analysis, including solutions with exponentially small terms.

Findings

Analytical expressions for far-field behavior and hump widths.

Relation of hump spacing to Lambert W function.

Efficient numerical tracking of solution branches.

Abstract

New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.