Date of Completion

6-24-2014

Embargo Period

6-24-2014

Keywords

Traveling wave solution; Traveling speed; Allen–Cahn equation; Fractional Laplacian; Continuation method; Hamiltonian identity

Major Advisor

Changfeng Gui

Associate Advisor

Yung-Sze Choi

Associate Advisor

Xiaodong Yan

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

We show the existence of traveling wave solutions to the Allen-Cahn equation with fractional Laplacians. A key ingredient is the estimation of the traveling speed of traveling wave solutions. In the meantime, we prove some qualitative properties of the solution, e.g., monotonicity, polynomial decays at infinity, Hamiltonian identity and Modica type estimates, and non-degeneracy. Moreover, we prove that for any balanced bistable nonlinearity, the traveling speeds linearly depend on the perturbation parameters.

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