KP-I equation: Difference between revisions

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The '''KP-I equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the
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>.
parameter <math>\lambda</math> is negative.


* Scaling is s1 + 2s2 + 1/2 = 0.  
* Scaling is <math>s_1 + 2s_2 + 1/2 = 0</math>.  
* GWP is known for data in a space roughly like (s1,s2) = (2,0), which is small in a certain weighted space [CoKnSt-p3].  Examples from [MlSauTz-p2] show that something like this type of additional condition is necessary.  
* 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 (2,0) intersect (-2,2) and no weight condition this is in [Kn-p]  
** 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 (s1,s2) = (3,0) intersect (-2,2) this is in [MlSauTz-p3].  
** 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]  
** 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 [MlSauTz-p3]; H^1 global weak solutions were constructed in [Tom1996].)  
* 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 (1,0) intersect (-1,1)) assuming also that yu is in L^2 [CoKeSt-p2].  Note that the latter property is preserved by the flow.  A technical refinement to Besov spaces is also available [CoKeSt-p2]; see also [CoKeSt-p3].  
* 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 (s1,s2) = (3/2+, 1/2+) this is in [MlSauTz-p2], 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 [MlSauTz-p2], [MlSauTz-p3]  
* 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 (3/2+) intersect (-1,1) is in [Kn-p].  
* A LWP result in a space roughly like <math>H^{3/2+,3/2+} \cap H^{-1,1}</math> is in [[Kn2004]].  
** For (s1,s2)=(2+,2+) this is in [IoNu1998]  
** For <math>H^{2+,2+}</math> this is in [[IoNu1998]]  
** For (s1,s2) = (3,3) this is in [IsMjStb1995], [Uk1989], [Sau1993]  
** For <math>H^{3,3}</math> 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.  
*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 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 [SauTz-p].  The corresponding problems for R x T and T x T remain open.  
*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 T x T one has LWP for (s1,s2) = (3,3) [IsMjStb1994]  
*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 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 <math>1 \leq k < 4/3</math> one has orbital stability [LiuWg1997], [BdSau1997].
*"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]]
[[Category:Equations]]

Latest revision as of 06:21, 21 July 2007

The KP-I equation is the special case of the Kadomtsev-Petviashvili equation when the parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} is negative. Well-posedness is usually studied in anisotropic Sobolev spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{s_1,s_2}({\Bbb R} \times {\Bbb R})} .

  • Scaling is .
  • GWP is known for data in a space roughly like , 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 and no weight condition this is in Kn2004
    • For data in a space which is roughly like this is in MlSauTz2002.
    • For small smooth data this was achieved by inverse scattering techniques in FsSng1992, Zx1990
  • On T, Global weak solutions were obtained for small data in Scz1987 and for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} data in Co1996. Assuming a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{3,0}} regularity at least, these global weak solutions are unique Scz1987. (The analogous uniqueness result on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Bbb R}} is in MlSauTz2002; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} global weak solutions were constructed in Tom1996.)
  • LWP in the energy space (which is essentially Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{1,0} \cap H^{-1,1}} ) assuming also that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^2} in standard Sobolev spaces MlSauTz2002b, MlSauTz2002
  • A LWP result in a space roughly like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{3/2+,3/2+} \cap H^{-1,1}} is in Kn2004.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{2+,2+}} this is in IoNu1998
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{3,3}} this is in IsMjStb1995, Uk1989, Sau1993
  • LWP and GWP in the energy space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{xxx}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{xxxxx}} ) then one has GWP in the energy space (when both the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} norm and Hamiltonian are finite) SauTz2000. This has been extended to the partly periodic case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Bbb T} \times {\Bbb R}} in SauTz2001. The corresponding problems for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Bbb R} \times {\Bbb T}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Bbb T} \times {\Bbb T}} remain open.
  • On Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Bbb T} \times {\Bbb T}} one has LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1 \leq k < 4/3} one has orbital stability LiuWg1997, BdSau1997.
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