For Tomcat put your application files directly in ~/appservers/apache-tomcat/webapps/ROOT directory. This can also be achieved by renaming your WAR to ROOT.war either locally before uploading or remotely on the server and restarting Tomcat. Another way of having your app at root URL is to modify default context in server.xml and set it like below.
<Host ... >
<Context path="" docBase="myapp" privileged="true" />
...
</Host>given your application is in ~/appservers/apache-tomcat/webapps/myapp directory.
Optionally map additional or all requests to your Tomcat using our custom Java Control Panel - JCP. You may want to make sure that you have welcome-file-list populated in your application's WEB-INF/web.xml.
For Jetty upload your application as root directory or root.war file to webapps directory.
For Glassfish check Host web application at Glassfish root URL
