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

TL;DR

This paper employs bifurcation theory for Fredholm operators to analyze periodic solutions of nonlocal pseudo-differential equations, including fractional Laplacians, and applies results to establish traveling wave existence in dispersive systems.

Contribution

It extends bifurcation analysis to a broad class of nonlocal operators, sharpening existing results for fractional Laplacians and deriving traveling wave solutions.

Findings

Existence of periodic solutions for a class of nonlocal equations.

Application to traveling wave solutions in dispersive equations.

Refinement of bifurcation results for fractional Laplacians.

Abstract

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 $\mathcal{L}u=\lambda u+|u|^{p}$, 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 $\mathcal{L}\equiv(-\Delta)^{s}$ 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.