[jQuery] manipulate the next element with a class

[Benn]

Is there a non-structure specific way of finding the next element with
a given class? for example you have a structure of:

<div id="container1">
  <a href="#" class="link1">link</a>
  <div class="a1">asdf</div>
</div><!-- #container1 -->

<div id="container2">
  <a href="#" class="link1">link</a>
  <div class="a1">ghjl</div>
</div><!-- #container2 -->

<div id="container3">
  <a href="#" class="link1">link</a>
  <div class="a1">qwer</div>
</div><!-- #container3 -->

When you click on the link in container1 the expected behavior is to
change the css on a1 in container2 but not in container1 or
container3. I have tried playing with parent, next and filter without
success. this is the best I have is:

$(document).ready(function(){
  $(".link1").click(function(){
    $(".a1").css({"visibility":"hidden"}); //hides all a1's but keeps
the space
  });
});