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== Dispersive PDE Wiki ==
Welcome to the '''Dispersive PDE Wiki'''!  These web pages are intended to present the latest results, [[:Category:Open problems|conjectures]], [[:Category:Bibliography|bibliography]], [[:Category:Concept|concepts]] and other material on the local and global [[well-posedness]] problems (and related questions) for non-linear [[dispersion relation|dispersive]] and [[wave equations]].  (We also have [[DispersiveWiki:About|a more detailed description of this wiki and its purpose]].)


Welcome to the Dispersive PDE Wiki!  These web pages are intended to present the latest results, conjectures, and other material
The information here has only recently been converted into wiki format, and there are still several rough edges to it.  We welcome updates, corrections, cleanup, and new contributions — just [[Special:Userlogin&type=signup|create an account]] and you're ready to edit!   
on the local and global [[well-posedness]] problem (and related questions) for non-linear dispersive and wave equations.  ([[DispersiveWiki:About|Click here]] for a more detailed description of this wiki and its purpose.)
If you are new to wikis, we have some [[Help:Contents|help files]] available.  If you are interested in contributing, you might want to visit our [[DispersiveWiki:Community Portal|community portal]] for current discussions and projects.


We welcome updates, corrections, and new contributions!  Please [[Help:Contents|click here]] for help on how to to do this.
== Classes of equations ==


{| style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%"
There are a [[:Category:Equations|large number]] of [[nonlinear]] [[dispersive]] and [[wave]] equations of interest, many of which arise from mathematical physics.  We can divide them into four classes:
| style="padding: .75pt .75pt .75pt .75pt" |
[[Purpose of this page]]
| style="padding: .75pt .75pt .75pt .75pt" |


| style="padding: .75pt .75pt .75pt .75pt" |
# [[Wave equations|Nonlinear wave equations]] ([[semilinear NLW|NLW]], [[semilinear NLW|NLKG]], [[Wave maps]], [[Maxwell-Klein-Gordon equations|Maxwell-Klein-Gordon]], [[Yang-Mills equations|Yang-Mills]], [[Einstein equations]], etc.)
[[Bibliography]]
# [[Schrodinger equations|Nonlinear Schrodinger equations]] ([[NLS equation|NLS]], [[Schrodinger maps|Schrodinger maps]], [[Cubic DNLS on R|DNLS]], [[Hartree equation|Hartree]], etc.)
|}
# [[KdV equations|Equations of Korteweg-de Vries type]] ([[Korteweg-de Vries equation|KdV]], [[Modified Korteweg-de Vries equation|mKdV]], [[Generalized Korteweg-de Vries equation|gKdV]], etc.)
# [[Other equations|Other equations and systems]] ([[Benjamin-Ono equation|Benjamin-Ono]], [[Kadomtsev-Petviashvili equation|Kadomtsev-Petviashvili]], [[Zakharov equation|Zakharov]], [[Davey-Stewartson system|Davey-Stewartson]], etc.)


<div class="MsoNormal" style="text-align: center"><center>
A complete list of equations discussed on this wiki [[:Category:Equations|can be found here]].
----
</center></div>


<center>The big three: Wave, Schrodinger, KdV</center>
== Other Web resources ==


{| style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
*Wikipedia has a [http://en.wikipedia.org/wiki/Partial_differential_equations partial differential equations page].
| style="padding: .75pt .75pt .75pt .75pt" |
*There is a [http://en.wikipedia.org/wiki/Category:Mathematics-related_lists list of math related pages on wikipedia] .
[[Wave equations]]
*Check out the [http://www.wikiwaves.org/index.php/Main_Page water waves wiki].
| style="padding: .75pt .75pt .75pt .75pt" |
[[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>

Latest revision as of 18:34, 10 June 2018

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

The information here has only recently been converted into wiki format, and there are still several rough edges to it. We welcome updates, corrections, cleanup, and new contributions — just create an account and you're ready to edit! If you are new to wikis, we have some help files available. If you are interested in contributing, you might want to visit our community portal for current discussions and projects.

Classes of equations

There are a large number of 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 equations, etc.)
  2. Nonlinear Schrodinger equations (NLS, Schrodinger maps, DNLS, Hartree, etc.)
  3. Equations of Korteweg-de Vries type (KdV, mKdV, gKdV, etc.)
  4. Other equations and systems (Benjamin-Ono, Kadomtsev-Petviashvili, Zakharov, Davey-Stewartson, etc.)

A complete list of equations discussed on this wiki can be found here.

Other Web resources

Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Main_Page&oldid=7576"