Take a look at – jQuery Plugin for Requesting Ajax-like File Downloads
The whole plugin
is just about 30 lines of code (including comments).
The call is fairly similar to jquery ajax call.
$.download('/export.php','filename=myPDF&format=pdf&content=" + pdfData );
Ofcourse, you have to set the content-type and Content-Disposition headers on the server side as you would for any such download.
In java I would do something like this
response.setContentType("application/pdf");
response.setHeader("Content-Disposition", "attachment; filename="exported.pdf");