Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective

In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation $\mathcal{L}u=\lambda u+\mathrm{\mid }u{\mathrm{\mid }}^p$, where $\mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $\lambda$ is the bifurcation parameter. Our general setting includes the fractional Laplacian $\mathcal{L}\equiv (-\mathrm{\Delta }{)}^s$ and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges.