KP-I equation

From DispersiveWiki

The KP-I equation is the special case of the Kadomtsev-Petviashvili equation when the parameter $λ$ is negative. Well-posedness is usually studied in anisotropic Sobolev spaces $H^{s_1,s_2}({\Bbb R} \times {\Bbb R})$ .

Scaling is $s 1 + 2 s 2 + 1 / 2 = 0$ .
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 ${\Bbb R}$ 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 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 $u x x x$ by $u x x x x x$ ) 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 ${\Bbb T} \times {\Bbb R}$ in SauTz2001. The corresponding problems for ${\Bbb R} \times {\Bbb T}$ and ${\Bbb T} \times {\Bbb T}$ remain open.
On ${\Bbb T} \times {\Bbb T}$ 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.