Well-posedness for the fifth order KP-II initial data problem in \(H^{s, 0}(\mathbb{R} \times \mathbb{T})\) (Q729944)

The authors consider the fifth order KP-II initial value problem \[ \begin{cases} \partial_tu -\partial_x^5 u+ \partial^{-1}_x \partial_y^2u +u\partial_xu = 0, \quad (t, x, y) \in (\mathbb{R}\times \mathbb{R}\times \mathbb{T}),\\ u(0, x, y) = u_0(x, y), \quad u_0 \in H^s (\mathbb{R}\times \mathbb{T}), \end{cases} \] where the operator \(\partial^{-1}x\) is defined by the Fourier multiplier \(i/\xi\) and \(\mathbb{T} = \mathbb{R}/(2\pi \mathbb{Z})\). Some well-posed properties for the initial value problem are obtained by using the Bourgain Fourier restriction method.