//Script to activate ajax
jQuery(document).ready(function($){
    var search_val=$("#s").val();
    $('#searchsubmit').click(function(){
        $.post(
            WPaAjax.ajaxurl,
            {
                action : 'wpa56343_search',
                search_val : search_val
            },
            function( response ) {
                $('#results').append( response );
            }
        );
    });

});

//function to setup wp_query
add_action('wp_ajax_wpa56343_search', 'wpa56343_search');
function wpa56343_search(){
    global $wp_query;
    $search = $_POST['search_val'];
    $args = array(
        's' => $search,
        'posts_per_page' => 5
    );
    $wp_query = new WP_Query( $args );

    get_template_part( 'search-results' );

    exit;
}

//html

      <div id="my_search">
       <form role="search" method="get" id="searchform" action="http://myurl.com/" >
        <input type="text" value="" name="s" id="s" />
        <input type="submit" id="searchsubmit" value="Search" />
       </form>
      </div>
      <div id="results"></div>