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 CoKnSt2001. Examples from MlSauTz2002b 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 Kn2004
  • For data in a space which is roughly like (s1,s2) = (3,0) intersect (-2,2) this is in MlSauTz2002.
  • 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 MlSauTz2002; 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 CoKnSt2003b. Note that the latter property is preserved by the flow. A technical refinement to Besov spaces is also available CoKnSt2003b; see also CoKnSt2001.
• For (s1,s2) = (3/2+, 1/2+) this is in MlSauTz2002b, 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 MlSauTz2002b, MlSauTz2002
• A LWP result in a space roughly like (3/2+) intersect (-1,1) is in Kn2004.
  • 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 SauTz2001. 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.