# KP-II equation

From DispersiveWiki

Jump to navigationJump to search
The **KP-II equation** is the special case of the Kadomtsev-Petviashvili equation when the
parameter is positive.

- Scaling is s1 + 2s2 + 1/2 = 0.
- GWP for s1 > -1/14, s2 = 0 IsMj2003.
- For s1 > -1/64 this is also in IsMj2001.

- GWP for s1 > -1/78, s2 = 0 Tk2000 assuming a moment condition.

LWP for s1 > -1/3, s2 = 0 TkTz2001, IsMj2001

- Weak solutions in a weighted L2 space were constructed in Fa1990.
- For s1 < -1/3 the natural bilinear estimate fails TkTz2001.
- Remark: Unlike KP-I, KP-II does not admit soliton solutions.

The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy + partial_zz), with s_1 regularity in the x direction and s_2 in the y,z directions. Scaling is now s_1 + 2s_2 – ½ = 0. In isotropic spaces, local well-posedness in H^s with s > 3/2 was established assuming the low frequency condition that partial_x^{-1} u is also in H^s Tz1999. Anisotropically, local well-posedness in the space s_1 > 1, s_2 > 0 was established in IsLopMj-p.