KP-I equation

The KP-I equation is the special case of the Kadomtsev-Petviashvili equation when the parameter $\lambda$ is negative.

• Scaling is $s1+2s2+1/2=0$.
• GWP is known for data in a space roughly like (s1,s2) = (2,0), which is small in a certain weighted space [CoKnSt-p3]. Examples from [MlSauTz-p2] show that something like this type of additional condition is necessary.
  • For data in a space roughly like (2,0) intersect (-2,2) and no weight condition this is in [Kn-p]
  • For data in a space which is roughly like (s1,s2) = (3,0) intersect (-2,2) this is in [MlSauTz-p3].
  • For small smooth data this was achieved by inverse scattering techniques in [FsSng1992], [Zx1990]
• On T, Global weak L2 solutions were obtained for small L2 data in [Scz1987] and for large L2 data in [Co1996]. Assuming a (3,0) regularity at least, these global weak solutions are unique [Scz1987]. (The analogous uniqueness result on R is in [MlSauTz-p3]; H^1 global weak solutions were constructed in [Tom1996].)
• LWP in the energy space (which is essentially (1,0) intersect (-1,1)) assuming also that yu is in L^2 [CoKeSt-p2]. Note that the latter property is preserved by the flow. A technical refinement to Besov spaces is also available [CoKeSt-p2]; see also [CoKeSt-p3].
• For (s1,s2) = (3/2+, 1/2+) this is in [MlSauTz-p2], however a certain technical condition at low frequencies has to be imposed (similarly for the results below). Note that without any such restriction the flow map is not even C^2 in standard Sobolev spaces [MlSauTz-p2], [MlSauTz-p3]
• A LWP result in a space roughly like (3/2+) intersect (-1,1) is in [Kn-p].
  • For (s1,s2)=(2+,2+) this is in [IoNu1998]
  • For (s1,s2) = (3,3) this is in [IsMjStb1995], [Uk1989], [Sau1993]
• LWP and GWP in the energy space ((1,0) intersect (-1,1)) without any localization condition is still an important unsolved problem.
• If one considers the fifth-order KP-I equation (replace uxxx by uxxxxx) then one has GWP in the energy space (when both the L2 norm and Hamiltonian are finite) [SauTz2000]. This has been extended to the partly periodic case (x,y) in T x R in [SauTz-p]. The corresponding problems for R x T and T x T remain open.
• On T x T one has LWP for (s1,s2) = (3,3) [IsMjStb1994]
• "Lump" soliton solutions exist for KP-I (and more generally for gKP-k-I for k = 1,2,3, but no solitons exist for k ³ 4 [WgAbSe1994], [Sau1993], [Sau1995], where solitons are understood to have at least some decay at infinity). When k > 4/3 these solitons are not orbitally stable [WgAbSe1994], [LiuWg1997], and in fact blowup solutions can be demonstrated to exist from a virial identity argument [Liu2001] (see also [TrFl1985], [Sau1993]). For 2 < k < 4 one in fact has strong orbital instability [Liu2001]. * For $1\leq k<4/3$ one has orbital stability [LiuWg1997], [BdSau1997].