It does not make any sense to introduce new operators just to save five characters. If you have to do this really often and/or find strpos unreadable, you can simply define a few helper functions:

    function contains($str1, $str2) {
        return false !== strpos($str1, $str2);
    }
    function startsWith($str1, $str2) {
        return 0 === strpos($str1, $str2);
    }
    function endsWith($str1, $str2) {
        return 0 === substr_compare($str1, $str2, -strlen($str2), strlen($str2));
    }

They have the additional advantage of being more clear than something like ~^.

You can also skip == and write equals($str, $str2) like Java does.
Helper functions will be slower, need to be defined somewhere, included, namespaced, tested, etc.
Operators are good to read and make things easier. Perl does a lot of similar things with =~ and people like it.