A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Cubillos, Pablo Granero-Belinchón, Rafael Sampedro, Juan Carlos |
| author_facet | Cubillos, Pablo Granero-Belinchón, Rafael Sampedro, Juan Carlos |
| contents | We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus,
\[
u_t+u u_x=Λ^{r}u-\varepsilon Λ^{s}u,\qquad x\in\mathbb T,
\] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global well-posedness for initial data in $H^{3}(\mathbb T)$. We then investigate the steady-state problem and show that the trivial branch undergoes bifurcation at the critical values $\varepsilon_k=k^{\,r-s}$, $k\in\mathbb N$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $(\varepsilon_k,0)$ and compute the bifurcation direction. To address the global continuation of these branches we derive global a priori bounds and apply a global alternative based on the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm maps of index zero. In particular, for the component bifurcating from the first critical point we prove that its $\varepsilon$-projection contains the interval $(2^{r-s},1)$, yielding the existence of nontrivial steady states for that parameter range. We complement the theory with numerical continuation results illustrating the bifurcation diagram and solution profiles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08107 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation Cubillos, Pablo Granero-Belinchón, Rafael Sampedro, Juan Carlos Analysis of PDEs 35B10, 35S15, 35S05, 47J15, 47A53 We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, \[ u_t+u u_x=Λ^{r}u-\varepsilon Λ^{s}u,\qquad x\in\mathbb T, \] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global well-posedness for initial data in $H^{3}(\mathbb T)$. We then investigate the steady-state problem and show that the trivial branch undergoes bifurcation at the critical values $\varepsilon_k=k^{\,r-s}$, $k\in\mathbb N$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $(\varepsilon_k,0)$ and compute the bifurcation direction. To address the global continuation of these branches we derive global a priori bounds and apply a global alternative based on the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm maps of index zero. In particular, for the component bifurcating from the first critical point we prove that its $\varepsilon$-projection contains the interval $(2^{r-s},1)$, yielding the existence of nontrivial steady states for that parameter range. We complement the theory with numerical continuation results illustrating the bifurcation diagram and solution profiles. |
| title | A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation |
| topic | Analysis of PDEs 35B10, 35S15, 35S05, 47J15, 47A53 |
| url | https://arxiv.org/abs/2602.08107 |