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- W3127879382 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 $textit{method of moving planes}$." @default.
- W3127879382 created "2021-02-15" @default.
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- W3127879382 date "2021-01-01" @default.
- W3127879382 modified "2023-09-23" @default.
- W3127879382 title "Symmetry of periodic waves for nonlocal dispersive equations" @default.
- W3127879382 doi "https://doi.org/10.5445/ir/1000129186" @default.
- W3127879382 hasPublicationYear "2021" @default.
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