Symmetry of Periodic Traveling Waves for Nonlocal Dispersive Equations

Abstract

Of concern is the a priori symmetry of traveling wave solutions of a general class of nonlocal dispersive equations , where is a Fourier multiplier operator with symbol . Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a large class of symbols guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. In contrast with the classically imposed setting in the water wave problem which assumes traveling waves to have a unique crest and trough per period or a monotone structure near troughs, we formulate a reflection criterion which does not presuppose a monotone structure on the wave profile. Thereby, the reflection criterion enables us to treat a priori solutions with multiple crests of different size per period. Moreover, our result applies not only to smooth traveling wave solutions, but also to those with singular crests around which some cancellation structure appears, including in particular waves with peaks or cusps. The proof relies on a so-called touching lemma, which is related to a strong maximum principle for elliptic operators, and a weak form of the celebrated method of moving planes.