Get search results without opening new page

I’m working on a simple derivative of TwentyTen theme.

I want to load search results into the currently open page using AJAX methods. To be able to do that, I need to obtain the output of the PHP code that does the search, without opening the new page.

Normally, after you type a search word, WordPress opens the new page with URL like this:

mydomain.com/?s=searchword

The output of the PHP code that does the searching is shown in the #container div of that page.

Question: how can I get the search results without opening the new page, so that I can load them using jQuery into the page that is currently open?

I would be grateful for your advice!