A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation

TL;DR

The paper analyzes the nonlocal Kuramoto--Sivashinsky equation on the one-dimensional torus to map the steady-state bifurcation structure. It establishes local well-posedness, identifies critical bifurcation points at $\varepsilon_k=k^{r-s}$, and constructs local Crandall--Rabinowitz branches with a computed subcritical direction. A global continuation is achieved via the Fitzpatrick--Pejsachowicz--Rabier degree, proving that the first bifurcation branch yields nontrivial steady states for $\varepsilon\in(2^{r-s},1)$ and describing asymptotic behavior as $\varepsilon\to0^+$. The work combines spectral analysis of the linearized operator, a priori estimates, and global bifurcation theory, complemented by numerical continuation that visualizes the bifurcation diagram and solution profiles.

Abstract

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.

Figures (2)

• Figure 1: Bifurcation diagram for $r=\tfrac{1}{2}$ and $s=\tfrac{3}{2}$.
• Figure 2: Profiles of some of the solutions (for $r=\tfrac{1}{2}$ and $s=\tfrac{3}{2}$) of the connected components $\mathscr{C}_{k}$ for $k=1,2,3,4$, respectively.

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof