<div class="modal" id="mkdocs_search_modal" tabindex="-1" role="dialog" aria-labelledby="searchModalLabel" aria-hidden="true">
    <div class="modal-dialog modal-lg">
        <div class="modal-content">
            <div class="modal-header">
                <h4 class="modal-title" id="searchModalLabel">Search</h4>
                <button type="button" class="close" data-dismiss="modal"><span aria-hidden="true">&times;</span><span class="sr-only">Close</span></button>
            </div>
            <div class="modal-body">
                <p>
                    From here you can search these documents. Enter
                    your search terms below.
                </p>
                <form>
                    <div class="form-group">
                        <input type="text" class="form-control" placeholder="Search..." id="mkdocs-search-query" title="Type search term here">
                    </div>
                </form>
                <div id="mkdocs-search-results"></div>
            </div>
            <div class="modal-footer">
            </div>
        </div>
    </div>
</div>

<div class="modal" id="mkdocs_search_modal" tabindex="-1" role="dialog" aria-labelledby="searchModalLabel" aria-hidden="true">
    <div class="modal-dialog modal-lg">
        <div class="modal-content">
            <div class="modal-header">
                <h4 class="modal-title" id="searchModalLabel">Search</h4>
                <button type="button" class="close" data-dismiss="modal"><span aria-hidden="true">&times;</span><span class="sr-only">Close</span></button>
            </div>
            <div class="modal-body">
                <p>
                    From here you can search these documents. Enter
                    your search terms below.
                </p>
                <form>
                    <div class="form-group">
                        <input type="search" class="form-control" placeholder="Search..." id="mkdocs-search-query" title="Type search term here">
                    </div>
                </form>
                <div id="mkdocs-search-results"></div>
            </div>
            <div class="modal-footer">
            </div>
        </div>
    </div>
</div>

Use type search for search input
Use type search for search input

doesn't make a difference to how it works but is more semantically correct.