Interaction of Ising-Bloch fronts with Dirichlet boundaries

Area of Science:

• Nonlinear dynamics
• Mathematical modeling
• Computational physics

Background:

• The Ising-Bloch bifurcation describes transitions in nonlinear systems.
• Spatial inhomogeneity and boundary conditions significantly impact system dynamics.
• Understanding front dynamics is crucial in various scientific fields.

Purpose of the Study:

• To investigate the Ising-Bloch bifurcation in the complex Ginzburg Landau equation (CGLE) and a FitzHugh Nagumo (FN) model.
• To analyze the influence of spatial inhomogeneity, specifically Dirichlet boundary conditions, on bifurcation dynamics.
• To establish the generality of the Ising-Bloch bifurcation by comparing its behavior in different systems.

Main Methods:

• Numerical simulations of the CGLE and FN models with Dirichlet boundary conditions.
• Derivation of reduced dynamical equations for the FN model to describe front dynamics.
• Analysis of front-boundary interactions and their dependence on system parameters.

Main Results:

• The interaction of fronts with boundaries is found to be similar in both the CGLE and FN models, confirming the generality of the Ising-Bloch bifurcation.
• Reduced dynamical equations for the FN model were successfully derived, providing insights into front behavior near boundaries.
• In the slow front (highly non-adiabatic) limit, front dynamics are governed by fixed points of these reduced equations located near the boundary.

Conclusions:

• The Ising-Bloch bifurcation is a general phenomenon applicable to diverse nonlinear systems, including the CGLE and FN models.
• Dirichlet boundary conditions introduce spatial inhomogeneity that leads to predictable front dynamics.
• Reduced dynamical equations offer a powerful tool for understanding and predicting front behavior in simplified, yet relevant, regimes.