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The '''KP-I equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the parameter <math>\lambda</math> is negative.  Well-posedness is usually studied in anisotropic Sobolev spaces <math>H^{s_1,s_2}({\Bbb R} \times {\Bbb R})</math>.

* Scaling is <math>s_1 + 2s_2 + 1/2 = 0</math>. 
* GWP is known for data in a space roughly like <math>H^{2,0}</math>, 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 <math>H^{2,0} \cap H^{-2,2}</math> and no weight condition this is in [[Kn2004]] 
** For data in a space which is roughly like <math>H^{3,0} \cap H^{-2,2}</math> this is in [[MlSauTz2002]]. 
** For small smooth data this was achieved by inverse scattering techniques in [[FsSng1992]], [[Zx1990]] 
* On T, Global weak <math>L^2</math> solutions were obtained for small <math>L^2</math> data in [[Scz1987]] and for large <math>L^2</math> data in [[Co1996]]. Assuming a <math>H^{3,0}</math> regularity at least, these global weak solutions are unique [[Scz1987]].  (The analogous uniqueness result on <math>{\Bbb R}</math> is in [[MlSauTz2002]]; <math>H^1</math> global weak solutions were constructed in [[Tom1996]].) 
* LWP in the energy space (which is essentially <math>H^{1,0} \cap H^{-1,1}</math>) assuming also that <math>yu \in L^2</math> [[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 <math>H^{3/2+,1/2+}</math> 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 <math>C^2</math> in standard Sobolev spaces [[MlSauTz2002b]], [[MlSauTz2002]] 
* A LWP result in a space roughly like <math>H^{3/2+,3/2+} \cap H^{-1,1}</math> is in [[Kn2004]]. 
** For <math>H^{2+,2+}</math> this is in [[IoNu1998]] 
** For <math>H^{3,3}</math> this is in [[IsMjStb1995]], [[Uk1989]], [[Sau1993]] 
*LWP and GWP in the energy space <math>H^{1,0} \cap H^{-1,1}</math> without any localization condition is still an important unsolved problem. 
*If one considers the fifth-order KP-I equation (replace <math>u_{xxx}</math> by <math>u_{xxxxx}</math>) then one has GWP in the energy space (when both the <math>L^2</math> norm and Hamiltonian are finite) [[SauTz2000]].  This has been extended to the partly periodic case <math>{\Bbb T} \times {\Bbb R}</math> in [[SauTz2001]].  The corresponding problems for <math>{\Bbb R} \times {\Bbb T}</math> and <math>{\Bbb T} \times {\Bbb T}</math> remain open. 
*On <math>{\Bbb T} \times {\Bbb T}</math> one has LWP for <math>(s_1,s_2) = (3,3)</math> [[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 <math>k \geq 4</math> [[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 <math>1 \leq k < 4/3</math> one has orbital stability [[LiuWg1997]], [[BdSau1997]].

[[Category:Integrability]]
[[Category:Equations]]