Main Page: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
(→‎Dispersive PDE Wiki: cleaned up intro a bit)
(→‎Dispersive PDE Wiki: Table of equations simplified)
Line 6: Line 6:
We welcome updates, corrections, and new contributions!  Please [[Help:Contents|click here]] for help on how to to do this.
We welcome updates, corrections, and new contributions!  Please [[Help:Contents|click here]] for help on how to to do this.


<div class="MsoNormal" style="text-align: center"><center>
== Classes of equations ==
----
</center></div>


<center>The big three: Wave, Schrodinger, KdV</center>
There are many nonlinear dispersive and wave equations of interest, many of which arise from mathematical physics.  We can divide them into four classes:


{| style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
# [[Wave equations|Nonlinear wave equations]] (NLW, NLKG, Wave maps, Maxwell-Klein-Gordon, Yang-Mills, Einstein, etc.)
| style="padding: .75pt .75pt .75pt .75pt" |
# [[Schrodinger equations|Nonlinear Schrodinger equations]] (NLS, Schrodinger maps, etc.)
[[Wave equations]]
# [[KdV equations|Equations of Korteweg-de Vries type]] (KdV, mKdV, gKdV, etc.)
| style="padding: .75pt .75pt .75pt .75pt" |
# [[Other equations]] (Kadomtsev-Petviashvhili, Zakharov, Davey-Stewartson, etc.)
[[Schrodinger equations]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[KdV equations]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[Wave estimates]]
 
* [[wave:Linear estimates]]
* [[wave:Bilinear estimates]]
* [[wave:Multilinear estimates]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[Schrodinger estimates]]
 
* [[schrodinger:Linear estimates]]
* [[schrodinger:Bilinear estimates]]
* [[schrodinger:Trilinear estimates]]
* [[schrodinger:Multilinear estimates]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:Airy estimates]]
 
* [[kdv:Linear estimates]]
* [[kdv:Bilinear estimates]]
* [[kdv:Trilinear estimates]]
* [[kdv:Multilinear estimates]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[wave:semilinear Semilinear NLW/NLKG]]
 
* [[wave:Scattering for NLW]]
* [[references:Scattering for NLKG]]
* [[wave:Sine-Gordon]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[schrodinger:Semilinear NLS]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:gKdV]]
 
* [[kdv:gKdV on R^+]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:nlw-2 Quadratic NLW/NLKG]
| style="padding: .75pt .75pt .75pt .75pt" |
[[schrodinger:Quadratic NLS]]
 
* [[schrodinger:Quadratic_NLS_on_R NLS-2 on R]
* [[schrodinger:Quartic_NLS_on_R^2 NLS-2 on R^2]]
* [[schrodinger:Quadratic_NLS_on_R^3 NLS-2 on R^3]]
* [[schrodinger:Quadratic_NLS_on_T NLS-2 on T]]
* [[schrodinger:Quadratic_NLS_on_T^2 NLS-2 on T^2]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:kdv KdV]] (gKdV-1)
 
* [[kdv:kdv on R KdV on R]]
* [[kdv:kdv on T KdV on T]]
* [[kdv:kdv on R+ KdV on R^+]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
Cubic NLW/NLKG
 
* [[wave:nlw-3 on R NLW-3 on R]]
* [[wave:nlw-3 on R^2 NLW-3 on R^2]]
* [[wave:nlw-3 on R^3 NLW-3 on R^3]]
| style="padding: .75pt .75pt .75pt .75pt" |
Cubic NLS
 
* [[schrodinger:Cubic_NLS_on_R NLS-3 on R]]
* [[schrodinger:Cubic_NLS_on_R^2 NLS-3 on R^2]]
* [[schrodinger:Cubic_NLS_on_R^3 NLS-3 on R^3]]
* [[schrodinger:Cubic_NLS_on_R^4 NLS-3 on R^4]]
* [[schrodinger:Cubic_NLS_on_T NLS-3 on T]]
* [[schrodinger:Cubic_NLS_on_T^2 NLS-3 on T^2]]
* [[schrodinger:Cubic_NLS_on_T^4 NLS-3 on T^4]]
* [[schrodinger:Cubic_NLS_on_RxT NLS-3 on R x T]]
* [[schrodinger:Cubic_NLS_on_S^6 NLS-3 on S^6]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:mkdv Modified KdV]] (gKdV-2)
 
* [[kdv:mKdV on R mKdV on R]]
* [[kdv:mKdV on T mKdV on T]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:nlw-4 Quartic NLW/NLKG]]
| style="padding: .75pt .75pt .75pt .75pt" |
Quartic NLS
 
* [[schrodinger:Quartic_NLS_on_R NLS-4 on R]]
* [[schrodinger:Quartic_NLS_on_T NLS-4 on T]]
* [[schrodinger:Quartic_NLS_on_R^2 NLS-4 on R^2]]
| style="padding: .75pt .75pt .75pt .75pt" |
gKdV-3
 
* [[kdv:gKdV_3 on R gKdV-3 on R]]
* [[kdv:gKdV_3 on T gKdV-3 on T]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
Quintic NLW/NLKG
 
* [[wave:nlw-5 on R NLW-5 on R]]
* [[wave:nlw-5 on R^2 NLW-5 on R^2]]
* [[wave:nlw-5 on R^3 NLW-5 on R^3]]
| style="padding: .75pt .75pt .75pt .75pt" |
Quintic NLS
 
* [[schrodinger:Quintic_NLS_on_R NLS-5 on R]]
* [[schrodinger:Quintic_NLS_on_R^2 NLS-5 on R^2]]
* [[schrodinger:Quintic_NLS_on_R^3 NLS-5 on R^3]]
* [[schrodinger:Quintic_NLS_on_T NLS-5 on T]]
| style="padding: .75pt .75pt .75pt .75pt" |
gKdV-4
 
* [[kdv:gKdV_4 on R gKdV-4 on R]]
* [[kdv:gKdV_4 on T gKdV-4 on T]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
Septic NLW/NLKG
 
* [[wave.htm#nlw-7 on R NLW-7 on R]]
* [[wave.htm#nlw-7 on R^2 NLW-7 on R^2]]
* [[wave.htm#nlw-7 on R^3 NLW-7 on R^3]]
| style="padding: .75pt .75pt .75pt .75pt" |
Septic NLS
 
* [[schrodinger:Septic_NLS_on_R NLS-7 on R]]
* [[schrodinger:Septic_NLS_on_R^2 NLS-7 on R^2]]
* [[schrodinger:Septic_NLS_on_R^3 NLS-7 on R^3]]
| style="padding: .75pt .75pt .75pt .75pt" |
 
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:dnlw D-NLW]]
 
* [[wave:gwp d-nlw GWP for D-NLW]]
* [[wave:gwp d-nlkg GWP for D-NLKG]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[schrodinger:d-nls D-NLS]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:hierarchy The KdV hierachy]]
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:dnlw-2 Quadratic DNLW]]
 
* [[wave:mkg MKG]]
** [[wave:mkg on R MKG on R]]
** [[wave:mkg on R^2 MKG on R^2]]
** [[wave:mkg on R^3 MKG on R^3]]
** [[wave:mkg on R^4 MKG on R^4]]
** [[wave:mkg on R^5+ MKG on R^5+]]
* [[wave:YM Yang Mills]]
** [[references:YM on R^2 YM on R^2]]
** [[wave:YM on R^3 YM on R^3]]
** [[wave:YM on R^4 YM on R^4]]
** [[wave:ym on R^5+ YM on R^5+]]
** [[wave:YMH on R^3 YMH on R^3]]
* [[wave:Maxwell-Dirac Maxwell-Dirac]]
* [[wave:Dirac-Klein-Gordon Dirac-Klein-Gordon]]
* [[wave:two-speed DNLW Two-speed DNLW-2]]
| style="padding: .75pt .75pt .75pt .75pt" |
Quadratic DNLS
 
* [[schrodinger:Maxwell-Schrodinger Maxwell-Schrodinger]]
| style="padding: .75pt .75pt .75pt .75pt" |
 
|-
| style="padding: .75pt .75pt .75pt .75pt" |
 
| style="padding: .75pt .75pt .75pt .75pt" |
Cubic DNLS
 
* [[schrodinger:dnls-3 on R DNLS-3 on R]]
| style="padding: .75pt .75pt .75pt .75pt" |
 
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:ddnlw DDNLW]]
 
* [[wave:wm Wave maps]]
** [[wave:wm on R WM on R]]
** [[wave:wm on R^2 WM on R^2]]
* [[wave:two-speed DDNLW Two-speed DDNLW]]
| style="padding: .75pt .75pt .75pt .75pt" |
DDNLS
 
* [[schrodinger:smaps Schrodinger Maps]]
| style="padding: .75pt .75pt .75pt .75pt" |
 
|-
| style="padding: .75pt .75pt .75pt .75pt" |
[[wave:Quasilinear Quasilinear NLW]]
 
* [[wave:gwp qnlw GWP for QNLW]]
* [[wave:Einstein Einstein equations]]
| style="padding: .75pt .75pt .75pt .75pt" |
Quasilinear NLS
| style="padding: .75pt .75pt .75pt .75pt" |
 
|-
| style="padding: .75pt .75pt .75pt .75pt" |
 
| style="padding: .75pt .75pt .75pt .75pt" |
[[schrodinger:Hartree Hartree equation]]
| style="padding: .75pt .75pt .75pt .75pt" |
[[kdv:Benjamin-Ono Benjamin-Ono equation]]
|}
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center>[[misc.html Other equations]]<nowiki>:</nowiki></center>
 
* Schrodinger-Ginsberg-Landau
* [[misc:Vlasov-Maxwell Vlasov-Maxwell]]
* [[misc:KP The Kadomtsev-Petvisasvhili equations]]
** [[misc:KP-I Kadomtsev-Petviashvili I]]
** [[misc:KP-II Kadomtsev-Petviashvili II]]
* [[misc:Zakharov Zakharov]]
** [[misc:Zakharov-1 Zakharov on R and T]]
** [[misc:Zakharov-2 Zakharov on R^2]]
** [[misc:Zakharov-3 Zakharov on R^3]]
** Klein-Gordon-Zakharov
** [[misc:WS Other Wave-Schrodinger systems]]
*** Ishimori
*** Davey-Stewartson
*** Yukawa
 
 
 
</div>

Revision as of 04:44, 27 July 2006

Dispersive PDE Wiki

Welcome to the Dispersive PDE Wiki! These web pages are intended to present the latest results, conjectures, bibliography and other material on the local and global well-posedness problem (and related questions) for non-linear dispersive and wave equations. (Click here for a more detailed description of this wiki and its purpose.)

We welcome updates, corrections, and new contributions! Please click here for help on how to to do this.

Classes of equations

There are many nonlinear dispersive and wave equations of interest, many of which arise from mathematical physics. We can divide them into four classes:

  1. Nonlinear wave equations (NLW, NLKG, Wave maps, Maxwell-Klein-Gordon, Yang-Mills, Einstein, etc.)
  2. Nonlinear Schrodinger equations (NLS, Schrodinger maps, etc.)
  3. Equations of Korteweg-de Vries type (KdV, mKdV, gKdV, etc.)
  4. Other equations (Kadomtsev-Petviashvhili, Zakharov, Davey-Stewartson, etc.)
Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Main_Page&oldid=1493"