Dynamical Systems and Traveling Waves in Almost Periodic Structures

AbstractThe paper is devoted to the study of traveling waves in almost periodic structures by applying dynamical system theory. Motivated by the fact that almost automorphic dynamics often exists (though almost periodic dynamics may not) in scalar almost periodic ODEs and parabolic PDEs, we introduce a definition of almost automorphic traveling waves and investigate the existence of such traveling waves in general almost periodic structures. In the author's earlier work, a notion of almost periodic traveling wave solution is introduced. Roughly, a solution is an almost automorphic (almost periodic) traveling wave solution if its propagating profile and speed are almost automorphic (almost periodic) functions. Our basic point of view is that traveling wave solutions are the limits of certain wave-like solutions. We therefore study the existence of almost automorphic traveling wave solutions through the long time behavior of wave-like solutions. We first investigate the existence of wave-like solutions. Then by utilizing dynamical system theory, we explore the “convergence” of wave-like solutions to alsmot automorphic (almost periodic) traveling wave solutions. Besides, based on the arguments established in the author's earlier work on bistable equations, we show the stability, uniqueness, and almost periodicity of almost automorphic traveling wave solutions in almost periodic equations of multi-stable type. Applying the general theory developed in the paper to almost periodic equations of KPP type, we show the existence of a family of almost automorphic traveling waves under some conditions