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Line periodic waves are solutions of the three-dimensional water wave problem which are periodic in one horizontal coordinate and do not depend on a second, transverse, horizontal coordinate. The transverse stability question is concerned with their stability with respect to three-dimensional perturbations, hence also depending on the horizontal coordinate in which the line periodic waves are constant. Relying upon an abstract, rather simple, linear instability criterion, we show that capillary-gravity periodic water waves are transversely unstable in several parameter regimes.

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Transverse Linear Instability of Line Periodic Traveling Waves for Water Wave Models