KP-II equation

From DispersiveWiki

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.
    • A similar result, with a slightly stricter constraint on s1 but no moment condition, was obtained in Tz2000.
    • 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 TkTz2001, IsMj2001

    • For s1 > -1/4, s2 = 0 this was shown in Tk2000b
    • 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 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.