KP-I equation

The KP-I equation is the special case of the Kadomtsev-Petviashvili equation when the parameter $\lambda$ is negative. Well-posedness is usually studied in anisotropic Sobolev spaces $H^{s_{1},s_{2}}({\mathbb {R}}\times {\mathbb {R}})$.

• Scaling is $s_{1}+2s_{2}+1/2=0$.
• GWP is known for data in a space roughly like $H^{2,0}$, 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 $H^{2,0}\cap H^{-2,2}$ and no weight condition this is in Kn2004
  • For data in a space which is roughly like $H^{3,0}\cap H^{-2,2}$ this is in MlSauTz2002.
  • For small smooth data this was achieved by inverse scattering techniques in FsSng1992, Zx1990
• On T, Global weak $L^{2}$ solutions were obtained for small $L^{2}$ data in Scz1987 and for large $L^{2}$ data in Co1996. Assuming a $H^{3,0}$ regularity at least, these global weak solutions are unique Scz1987. (The analogous uniqueness result on ${\mathbb {R}}$ is in MlSauTz2002; $H^{1}$ global weak solutions were constructed in Tom1996.)
• LWP in the energy space (which is essentially $H^{1,0}\cap H^{-1,1}$) assuming also that $yu\in L^{2}$ 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 $H^{3/2+,1/2+}$ 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 $H^{3/2+,3/2+}\cap H^{-1,1}$ is in Kn2004.
  • For $H^{2+,2+}$ this is in IoNu1998
  • For $H^{3,3}$ this is in IsMjStb1995, Uk1989, Sau1993
• LWP and GWP in the energy space $H^{1,0}\cap H^{-1,1}$ without any localization condition is still an important unsolved problem.
• If one considers the fifth-order KP-I equation (replace $u_{xxx}$ by $u_{xxxxx}$) then one has GWP in the energy space (when both the $L^{2}$ norm and Hamiltonian are finite) SauTz2000. This has been extended to the partly periodic case ${\mathbb {T}}\times {\mathbb {R}}$ in SauTz2001. The corresponding problems for ${\mathbb {R}}\times {\mathbb {T}}$ and ${\mathbb {T}}\times {\mathbb {T}}$ remain open.
• On ${\mathbb {T}}\times {\mathbb {T}}$ one has LWP for $(s_{1},s_{2})=(3,3)$ 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 $k\geq 4$ 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.