Center Stable Manifolds Around Line Solitary Waves of the Zakharov–Kuznetsov Equation

In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on \({\mathbb {R}} \times {\mathbb {T}}_L\) and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space.