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

Juan Carlos Sampedro

TL;DR

The paper develops a unified bifurcation framework for nonlocal periodic equations of the form $\mathcal{L}u=\lambda u+|u|^p$ on the 1D torus, where $\mathcal{L}$ is a Fourier-multiplier operator including the fractional Laplacian. Using Crandall–Rabinowitz local bifurcation and a global degree-theoretic alternative for Fredholm index-zero operators, it proves the existence of nontrivial even periodic solutions bifurcating from each spectral mode and analyzes their global continua; in particular, for $\mathcal{L}=(-\Delta)^s$, the continua are global and their $\lambda$-projections cover spectral intervals $(k^{2s},(k+1)^{2s})$. The work also establishes sharp a priori bounds, proves specialized results for the fractional Laplacian, and treats the Benjamin–Ono equation, where explicit traveling waves are recovered and connected to the bifurcation diagram. Overall, it provides a rigorous, operator-theoretic approach to nonlocal periodic traveling waves and their global bifurcation structure. Its framework broadens the scope of nonlocal wave analysis beyond classical local operators, with implications for dispersive models and periodic solutions in nonlocal media.

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 $\mathcal{L}u=λu+|u|^{p}$, where $\mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $λ$ is the bifurcation parameter. Our general setting includes the fractional Laplacian $\mathcal{L}\equiv(-Δ)^{s}$ 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.

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

TL;DR

The paper develops a unified bifurcation framework for nonlocal periodic equations of the form on the 1D torus, where is a Fourier-multiplier operator including the fractional Laplacian. Using Crandall–Rabinowitz local bifurcation and a global degree-theoretic alternative for Fredholm index-zero operators, it proves the existence of nontrivial even periodic solutions bifurcating from each spectral mode and analyzes their global continua; in particular, for , the continua are global and their -projections cover spectral intervals . The work also establishes sharp a priori bounds, proves specialized results for the fractional Laplacian, and treats the Benjamin–Ono equation, where explicit traveling waves are recovered and connected to the bifurcation diagram. Overall, it provides a rigorous, operator-theoretic approach to nonlocal periodic traveling waves and their global bifurcation structure. Its framework broadens the scope of nonlocal wave analysis beyond classical local operators, with implications for dispersive models and periodic solutions in nonlocal media.

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 , where is a nonlocal pseudodifferential operator defined as a Fourier multiplier and is the bifurcation parameter. Our general setting includes the fractional Laplacian 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.
Paper Structure (14 sections, 41 theorems, 292 equations, 3 figures)

This paper contains 14 sections, 41 theorems, 292 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Exposition of the problem and Nonlinear operators
  3. Spectral theory for the linearization
  4. Local bifurcation
  5. Constant solutions
  6. A priori bounds
  7. A priori bounds for the fractional Laplacian
  8. Global bifurcation theory
  9. Global bifurcation for the fractional laplacian.
  10. Benjamin--Ono Equation
  11. Preliminaries on nonlinear spectral theory
  12. Preliminaries on Bifurcation Theory
  13. The Crandall--Rabinowitz theorem
  14. Global bifurcation theory

Key Result

Theorem 1.1

Suppose that $s\geq \tfrac{1}{2}$ and $2\leq p<4s+1$. Then, the pseudo-differential equation admits at least one non-constant even solution for every $\lambda\in (\mathfrak{m}(1),2^{2s}\mathfrak{m}(2))$. Moreover, if $p=2$, equation Eq111 admits one non-constant even solution for every

Figures (3)

  • Figure 1: Graphic representation of the connected subset $A$
  • Figure 2: Illustration of the connected component $\mathscr{C}_{0}$ and $\mathscr{C}_{1}$ and the action of the homeomorphism $T$
  • Figure 3: Bifurcation diagram of equation \ref{['rr']}

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 62 more