Asymptotic analysis and numerical simulation of traveling wave solutions for crystal growth with corner regularization

We investigate the growth of crystal shape in the presence of corners. The evolution equation can be reduced to a second order partial differential equation. But near the corner it is backward parabolic so a fourth order regularizing term is added to make the interface evolution well-posed in the whole domain. We prove that the dynamical behavior of a single regularized corner can be described by a traveling wave solution. From this we derive the traveling wave speed in the limit of vanishing regularization. We rigorously prove the existence of such traveling waves. When there are multiple corners, we study a reduced dynamics which connects the traveling waves and parabolic “heat” equations. In addition, the dynamics of a wrinkling shape is studied leading to the concept and precise formulation of pulsating waves. Their asymptotic speed is also investigated. Numerical simulations using spectral methods are presented to visualize the kink structure dynamics and the traveling and pulsating wave speeds