Bifurcations of front motion in passive and active Allen-Cahn-type equations

Abstract

The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occuring different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyse fronts for more involved AC-type models. In particular, we consider a cubic-quintic variational AC model and two different nonvariational generalisations. We determine and compare the bifurcation diagrams of front solutions in the four considered models.

References in corpus (17)

• Motility-Induced Phase Separation
• Front propagation into unstable states
• Phase transition in the collective migration of tissue cells: experiment and model
• A continuum theory of phase separation kinetics for active Brownian particles
• Scalar ${\varphi }^4$ field theory for active-particle phase separation
• Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview
• Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria
• Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
• Traveling and resting crystals in active systems
• Structural Stability and Renormalization Group for Propagating Fronts
• Active crystals and their stability
• Resting and Traveling Localized States in an Active Phase-Field-Crystal Model
• Systematic extension of the Cahn-Hilliard model for motility-induced phase separation
• Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen-Cahn Equation
• Phase separation patterns from directional quenching
• Bifurcations of front motion in passive and active Allen-Cahn-type equations
• Effect of the orientational relaxation on the collective motion of patterns formed by self-propelled particles

Cited by in corpus (7)

• Classical dynamical density functional theory: from fundamentals to applications
• Phase-field crystal description of active crystallites: elastic and inelastic collisions
• Bifurcations of front motion in passive and active Allen-Cahn-type equations
• Demixing of two species via reciprocally concentration-dependent diffusivity
• Steady states and coarsening in one-dimensional driven Allen-Cahn system
• Hyperparameter Optimization in the Estimation of PDE and Delay-PDE models from data
• Instability in large bounded domains -- branched versus unbranched resonances