Given a comment you made on another user's post, try the following:
var $li = jQuery(this).parents('li').get(0);
var ajaxName = $li.className.match(/(?:^|\s)ajax-(.*?)(?:$|\s)/)[1];
Demo can be found here
Below kept for reference only
var ajaxName = 'ajax-first'.match(/(\w+)$/)[0];
alert(ajaxName);
Use the \w (word) pattern and bind it to the end of the string. This will force a grab of everything past the last hyphen (assuming the value consists of only [upper/lower]case letters, numbers or an underscore).
The non-regex approach could also use the String.split method, coupled with Array.pop.
var parts = 'ajax-first'.split('-');
var ajaxName = parts.pop();
alert(ajaxName);