Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

Abstract:

We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space $H^{s_1,s_2}(\mathbb {R}^2)$ with $s_1>-\frac 12$ and $s_2\geq 0$. On the $H^{s_1,0}(\mathbb {R}^2)$ scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation: \[ (u_t - |D_x|^\alpha u_x + (u^2)_x)_x + u_{yy} = 0, \quad u(0) = u_0, \] for $\frac 43<\alpha \leq 6$, $s_1>\max (1-\frac 34 \alpha ,\frac 14-\frac 38 \alpha )$, $s_2\geq 0$ and $u_0\in H^{s_1,s_2}(\mathbb {R}^2)$. We deduce global well-posedness for $s_1\geq 0$, $s_2=0$ and real valued initial data.