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arXiv · 2101.02269

Green's function for the fractional KdV equation on the periodic domain via Mittag-Leffler's function

Uyen Le·Dmitry E. Pelinovsky
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Abstract

The linear operator $c + (-Δ)^{α/2}$, where $c > 0$ and $(-Δ)^{α/2}$ is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg--de Vries equation. We establish a relation of the Green's function of this linear operator with the Mittag--Leffler function, which was previously used in the context of Riemann--Liouville's and Caputo's fractional derivatives. By using this relation, we prove that Green's function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every $c > 0$ and every $α\in (0,2]$. On the other hand, we argue from numerical approximations that in the case of $α\in (2,4]$, the Green's function is positive and single-lobe for small $c$ and non-positive and non-single lobe for large $c$.

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Uyen Le, Dmitry E. Pelinovsky. 2021-11-05. Green's function for the fractional KdV equation on the periodic domain via Mittag-Leffler's function. https://doi.org/10.1515/fca-2021-0063

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