Sixth order modification of the Cahn-Hilliard equation

TL;DR

The paper investigates a sixth-order modification of the convective-viscous Cahn–Hilliard equation that arises from a nonconstant gradient coefficient and a (Delta w)^2 term in the thermodynamic potential. By nondimensionalizing, the authors obtain a sixth-order PDE for the order parameter u and construct exact static kink and traveling-wave solutions under specific parameter constraints, highlighting a balance between driving and dissipation. They provide explicit expressions for wave speed and amplitude, identify the admissible parameter domain, and show how higher-order derivatives soften fronts and modify dynamics, extending previous analyses of fourth-order CH models. The results have implications for phase-field modeling of inhomogeneous systems where gradient terms are highly influential.

Abstract

We consider the sixth-order convective-viscous Cahn-Hilliard equation, different from the standard fourth-order Cahn-Hilliard equation due to the modified expression for the thermodynamic potential. In such modified thermodynamic potential the coefficient at the square gradient term is order-parameter-dependent. It also contains the square of the Laplacian. This results in a sixth-order differential equation and additional nonlinear terms in the equation. We obtained several exact static- and traveling wave solutions and studied the dependence of solutions on the parameters of the system.

Figures (1)

• Figure 1: Illustration of the condition \ref{['3.14']}: the allowed domain in the $\left(\frac{\rho }{\theta } ,\, \lambda \right)$ plane is hatched in blue.