Symmetry of periodic waves for nonlocal dispersive equations

Of concern is the $\textit{a priori}$ symmetry of traveling wave solutions for a general class of nonlocal dispersive equations
$$u_t + (u^2 + Lu)_x = 0,$$ where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols m guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formulate a $\textit{reflection criterion}$, which allows to affirm the symmetry of periodic traveling waves. The reflection criterion weakens the assumption of monotonicity between trough and crest and enables to treat $\textit{a priori}$ solutions with multiple crests of different sizes per period. Moreover, our result not only applies to smooth solutions, but also to traveling waves with a non-smooth structure such as peaks or cusps at a crest. The proof relies on a so-called $\textit{touching lemma}$, which is related to a strong maximum principle for elliptic operators, and a weak form of the celebrated $\textit{method of moving planes}$.