The best I can tell it's a div themed with an input element. That being said, you can probably do one of two things:
Bind to the input's wrapper such as:
$('__INPUT_SELECTOR__') // grab the input field
.closest('div') // traverse up to the parent div
.bind('click',function(){
// grab the click event and make it a form submit
$(this).closest('form').submit();
});
OR, you go the other direction:
$('div.ui-input-search') // find the div first
.bind('click',function(){
// now you have a click event on the div. Transfer it to the form's submit
$(this).closest('form').submit();
});
Just depends the direction you want to go. (And probably should be using the jquerymobile events, not classic jQuery ones (so you can catch figner taps, etc.)) Also, this should be interpreted as pseudo-code as I am not familiar with the library, just going with what I know.
EDIT
And so it's clear, I realize more has to be done (in terms of checking) for the DIV's click event. You would need to test the click was actually performed on the DIV not the input which results in a DIV event fire.
Just throwing it out there to avoid "this doesn't work" comments. This is why it's pseudo-code.