A very limited list of open source wiki engines that can run on a LAMP (using MySQL and PHP) :

• CockatooWiki (http://www.moldycheese.biz/cockatoowiki.html) -- is simple PhpMySQL-based wiki solution. Aims toward easy setup and customization.
• CoTeia (http://freshmeat.net/projects/coteia/) -- XML-based Brazilian collaborative authoring tool with access control (optional), concurrency control (WebDAV), chat server, annotation server and site map. Runs on MySQL.
• coWiki (http://www.develnet.org) -- follows the tradition of loose wikis with easy and intuitional markup, adding Unixlike access management, a directory/document hierarchy, and a plugin API for your functionalities and enhancements. All documents are parsed to XML for further export and transformation. coWiki is modular, template-based and multilingual. Requires PHP5 and MySQL
• ErfurtWiki (http://erfurtwiki.sourceforge.net/) -- embeddable into existing sites, uses SQL or flat-file backend, single script, allows plugins.
• MediaWiki -- The same engine used by WikiMedia projects: Wikipedia, Wiktionary, Wikiquote, WikiBooks etc. PHP+MySQL, GPL.
• PhpWiki -- the PHP Wiki with all possible backends
• WakkaWiki -- very light PHP/MySQL Wiki with quick installation, access control through per-page access control lists, full revision control and page comments. Development of WakkaWiki has stopped. The following forks are available:
  • UniWakka [1] (http://uniwakka.sourceforge.net/) a fork of Wakka for the scientific/academic community.
  • WackoWiki [2] (http://wackowiki.com/WackoWiki) is a fork of Wakka, with many new features (multilingual).
  • WikkaWiki [3] (http://wikka.jsnx.com) a Wakka-fork designed for lightweightness and easy extensibility.
• WikiTypeFramework (http://wtfw.sourceforge.net/) -- PHP/MySQL collaborative Web development system. WTF is not a Wiki engine, although it can be used as one. It is actually a generic web content management engine allowing you to create web documents easily from the comfort of your browser. If you must compare it to something, consider it a cross between a Blog, a Wiki, and a forum.
• wikiX (http://wikix.sourceforge.net/) -- PHP/MySQL-based wiki. It aims to allow users to redefine syntax by using wikiXmacros and various stages of PHP plugin.
• WikkiTikkiTavi (http://tavi.sourceforge.net/)