# Symmetry of periodic waves for nonlocal dispersive equations

Bruell, Gabriele; Pei, Long

##### Abstract:
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 $\textit{method of moving planes}$.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsmonat/-jahr 02.2021 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000129186 Verlag Karlsruher Institut für Technologie (KIT) Umfang 16 S. Serie CRC 1173 Preprint ; 2021/5 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Externe Relationen Siehe auch Schlagwörter traveling waves, symmetry, nonlocal dispersive equation, methods of moving planes
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