In this paper, we construct center stable manifolds around unstable line solitary waves of the Zakharov–Kuznetsov equation on two dimensional cylindrical spaces $ \mathbb {R} \times \mathbb {T}_L $ ($ {\mathbb T}_L = {\mathbb R}/2\pi L {\mathbb Z} $). In the paper [39], center stable manifolds around unstable line solitary waves have been constructed excluding the case of critical speeds $ c \in \{4n^2/5L^2;n \in {\mathbb Z}, n>1\} $. In the case of critical speeds $ c $, any neighborhood of the line solitary wave with speed $ c $ in the energy space includes solitary waves which are depend on the direction $ {\mathbb T}_L $. To construct center stable manifolds including their solitary waves and to recover the degeneracy of the linearized operator around line solitary waves with critical speed, we prove the stability condition of the center stable manifold for critical speed by applying to the estimate of the 4th order term of a Lyapunov function in [37] and [38].

中文翻译：

---

以临界速度围绕Zakharov–Kuznetsov方程的线孤立波将稳定流形居中

在本文中，我们围绕二维圆柱空间$ \ mathbb {R} \ times \ mathbb {T} _L $（$ {\\ mathbb T} _L = { \ mathbb R} / 2 \ pi L {\ mathbb Z} $）。在论文[39），除了临界速度$ c \ in \ {4n ^ 2 / 5L ^ 2; n \ in {\ mathbb Z}，n> 1 \} $之外，已经构造了围绕不稳定线孤波的中心稳定流形。在临界速度$ c $的情况下，在能量空间中速度为$ c $的线孤波的任何邻域都包含孤波，这些孤波取决于方向$ {\ mathbb T} _L $。为了构造包括孤波的中心稳定歧管并恢复具有临界速度的线孤波在临界速度附近的线性化算子的简并性，我们通过将方程的四阶项的估计应用于证明中心稳定歧管对于临界速度的稳定性条件。 [37] 和 [38]。