Computation of Minimum Energy Paths for Quasi-Linear Problems

Abstract

We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E., Ren, and Vanden-Eijnden (J. Chem. Phys., vol. 126 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional "ridges" of saddle-continua. We then apply the same method to locate saddle solutions and transition paths of the two-dimensional quasi-linear problem. The method developed is applicable to a general class of quasi-linear PDEs.

A reformulated Krein matrix for star-even polynomial operators with applications

“…In this section, we show numerical results to illustrate the theoretical results of the previous section. First, we can construct a primary pulse solution U (x) numerically using the string method from [4]. The top two panels of Figure 5 show these solutions for the same values of c as in [6, Figure 3].…”