KP-II equation

The KP-II equation is the special case of the Kadmotsev-Petviashvili equation when the parameter $\lambda$ 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 [Tk-p] assuming a moment condition.
  • A similar result, with a slightly stricter constraint on s1 but no moment condition, was obtained in [Tz-p].
  • For s1 = s2 ³ 0 this was proven in [Bo1993c], and this argument also applies to the periodic setting. Heuristically this result is indicated by the local smoothing estimates in [Sau1993].

LWP for s1 > -1/3, s2 = 0 [TkTz-p4], [IsMj2001]

• • For s1 > -1/4, s2 = 0 this was shown in [Tk-p2]
  • For s1 > -e, s2 = 0 and small data this was shown in [Tz1999].
  • For s1 = s2 ³ 0 this was proven in [Bo1993c], and this argument also applies to the periodic setting.
  • For s1, s2 ³ 3 this is in [Uk1989]
  • Related results are in [IoNu1998], [IsMjStb2001].
• Weak solutions in a weighted L2 space were constructed in [Fa1990].
• For s1 < -1/3 the natural bilinear estimate fails [TkTz-p4].
• 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].