Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909971321454592 |
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| author | Sampedro, Juan Carlos |
| author_facet | Sampedro, Juan Carlos |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective Sampedro, Juan Carlos Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35S15, 35J61, 35S30, 35S05 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. |
| title | Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35S15, 35J61, 35S30, 35S05 |
| url | https://arxiv.org/abs/2409.04253 |