Traveling Waves in Time Almost Periodic Structures Governed by Bistable Nonlinearities: I. Stability and Uniqueness

The current series of two papers is concerned with traveling wave solutions of time almost periodic nonlinear evolution equations. In the first of the series, we introduce a definition of almost periodic traveling wave solutions and establish the stability and uniqueness of such solutions to time almost periodic bistable nonlinear equations with one dimensional continuous space variable. We also investigate the stability and uniqueness of such solutions to time almost periodic bistable equations with one dimensional discrete space variable. In the second of the series, we will study the existence of almost periodic traveling wave solutions to time almost periodic bistable equations with continuous as well as discrete space variables.