Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations

Abstract

In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior.

Keywords

• wave-front propagation
• Allen–Cahn equation
• polynomial order in double-well potentials