Hi,
You can will need to customize much on the search module to achieve that.
1) Update the search display in ‘topbar’ block: root/templates/ja_platon/tpls/blocks/topbar.php
<?php if ($this->countModules('head-search')) : ?>
<!-- HEAD SEARCH -->
<div class="dropdown nav-search pull-right<?php $this->_c('head-search') ?>">
<a data-toggle="dropdown" href="#" class="dropdown-toggle">
<i class="fa fa-search"></i>
</a>
<div class="nav-child dropdown-menu container">
<div class="dropdown-menu-inner">
<jdoc:include type="modules" name="<?php $this->_p('head-search') ?>" style="raw" />
</div>
</div>
</div>
<!-- //HEAD SEARCH -->
<?php endif ?>
2) Customize style for this search module.
Regards