# KP-I equation

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

• Scaling is ${\displaystyle s_{1}+2s_{2}+1/2=0}$.
• GWP is known for data in a space roughly like ${\displaystyle H^{2,0}}$, 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 ${\displaystyle H^{2,0}\cap H^{-2,2}}$ and no weight condition this is in Kn2004
• For data in a space which is roughly like ${\displaystyle H^{3,0}\cap H^{-2,2}}$ this is in MlSauTz2002.
• For small smooth data this was achieved by inverse scattering techniques in FsSng1992, Zx1990
• On T, Global weak ${\displaystyle L^{2}}$ solutions were obtained for small ${\displaystyle L^{2}}$ data in Scz1987 and for large ${\displaystyle L^{2}}$ data in Co1996. Assuming a ${\displaystyle H^{3,0}}$ regularity at least, these global weak solutions are unique Scz1987. (The analogous uniqueness result on ${\displaystyle {\mathbb {R}}}$ is in MlSauTz2002; ${\displaystyle H^{1}}$ global weak solutions were constructed in Tom1996.)
• LWP in the energy space (which is essentially ${\displaystyle H^{1,0}\cap H^{-1,1}}$) assuming also that ${\displaystyle yu\in L^{2}}$ 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 ${\displaystyle H^{3/2+,1/2+}}$ 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 ${\displaystyle C^{2}}$ in standard Sobolev spaces MlSauTz2002b, MlSauTz2002
• A LWP result in a space roughly like ${\displaystyle H^{3/2+,3/2+}\cap H^{-1,1}}$ is in Kn2004.
• For ${\displaystyle H^{2+,2+}}$ this is in IoNu1998
• For ${\displaystyle H^{3,3}}$ this is in IsMjStb1995, Uk1989, Sau1993
• LWP and GWP in the energy space ${\displaystyle H^{1,0}\cap H^{-1,1}}$ without any localization condition is still an important unsolved problem.
• If one considers the fifth-order KP-I equation (replace ${\displaystyle u_{xxx}}$ by ${\displaystyle u_{xxxxx}}$) then one has GWP in the energy space (when both the ${\displaystyle L^{2}}$ norm and Hamiltonian are finite) SauTz2000. This has been extended to the partly periodic case ${\displaystyle {\mathbb {T}}\times {\mathbb {R}}}$ in SauTz2001. The corresponding problems for ${\displaystyle {\mathbb {R}}\times {\mathbb {T}}}$ and ${\displaystyle {\mathbb {T}}\times {\mathbb {T}}}$ remain open.
• On ${\displaystyle {\mathbb {T}}\times {\mathbb {T}}}$ one has LWP for ${\displaystyle (s_{1},s_{2})=(3,3)}$ 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 ${\displaystyle k\geq 4}$ 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 ${\displaystyle 1\leq k<4/3}$ one has orbital stability LiuWg1997, BdSau1997.