Existence of traveling waves for vector valued gradient flows
Xinfu Chen, Zhilei Liang
TL;DR
This paper proves the existence of traveling wave solutions for vector-valued Allen-Cahn equations within Ginzburg-Landau models, identifying bounds for the maximum wave speed using a variation method applicable to gradient flow systems.
Contribution
It establishes the existence of traveling waves for vector-valued Allen-Cahn equations and determines bounds for the largest wave speed using a novel variation technique.
Findings
Existence of traveling waves for vector-valued Allen-Cahn equations.
Bounds for the largest wave speed are provided.
Method applicable to gradient flow systems.
Abstract
Allen-Cahn equation is a fundamental continuum model that describes phase transitions in multi-component mixtures. We prove the existence of traveling waves for vector valued Allen-Cahn equations in the context of Ginzburg-Landau theories; in addition, we find the largest wave speed and provide its bounds from upper and below. Our method is based on a variation technique and can be applied to system of equations with a gradient flow structure.
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Taxonomy
TopicsSolidification and crystal growth phenomena · Nonlinear Partial Differential Equations · Advanced Mathematical Modeling in Engineering