This might be a great place to start as you can compare different wiki applications and make comparisons to what you need:
http://www.wikimatrix.org/

There seems to be a lot of options, but some are discontinued.
[spoiler] 3F Wiki
@Wiki
Anwiki
bitweaver
bLADE Wiki
BoltWire
BrainKeeper
BusinessWiki
ButorWiki
CanvasWiki
CentralDesktop
Clearspace
ClearWiki
codeBeamer
Confluence
ConnectedText
Corendal Wiki
Cospire
Daisy
DokuWiki
Drupal Wiki
EditMe
ErfurtWiki
FlexWiki
Foswiki
Friki
Gazest
GeboGebo
GeniusWiki
Giki
Git-Wiki
gitit
Groupswiki
Hatta
IkeWiki
ikiwiki
Incentive
Instiki
Intodit
Jacwiki
JAMWiki
JaWiki
JOTWiki
JSPWiki
KeheiWiki
KWikiKWiki
LionWiki
Luminotes
LunaWiki
MediaWiki
Metadot Wiki
MicKI
Midgard Wiki
MindTouch
miniWiki
MoinMoin
MojoMojo
MoniWiki
Netcipia
nexdo
Oddmuse
OpenWikiNG
PageWork
PAUX
PBwiki
Personal PPC Wiki
PhpWiki
PicoWiki
Pier
Pimki
PmWiki
PodWiki
Portili Team Wiki
ProjectForum
ProntoWiki
ProWiki
PukiWiki
Riki
SamePage
ScrewTurn Wiki
SharePoint Wiki Plus
Socialtext
Springnote
Sputnik
SubWiki
Swirrl
Sycamore
telepark.wiki
ThoughtFarmer
TiddlyWiki
Tiki Wiki CMS Groupware
Traction TeamPage
TracWiki
Triki-Wiki
TWiki
UniWakka
UseMod
VQWiki
WackoWiki
Wagn
Wala Wiki
Wetpaint
Wicked
Wiclear
WiGit
Wikepage
Wiki CSharp .NET
Wiki on a Stick
Wiki Spot
Wiki-Toolkit
Wikia
WikiASP
WikiCrowd
WikiDoc
Wikidot
WikiNi
WikiSH
Wikispaces
WikkaWiki
WikyBlog
wxWikiServer
XoWiki
XWiki
yawiki
Zoho Wiki
Zwiki [/spoiler]

If you know anything about wiki applications, what's good and what's not, please give us your insight.


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Zwiki was most compare with the following: PhpWiki (37%), FlexWiki (16%), Wikispaces (16%) → Compare


Here's more info on PHPWiki: http://www.wikimatrix.org/show/PhpWiki

If you own the host access to your website, many times they offer Wiki Capabilities.