Create a field in the managed bean to hold the default value when navigating:
private String filter = "car"; // + getter/setter
Then, in the view:
<h:form>
<h:inputText label="Type search filter :" value="#{movie.filter}" />
<h:dataTable value="#{movie.getSearchMovieList(movie.filter)}" var="c" ... >
...
</h:dataTable>
<h:commandButton value="Search" /> // no action attribute means refreshing
<h:form>