<input type="text" name="fullname" id="fullname" placeholder="Full name">

<button onclick="setSession()"> GO </button>

<script>
function setSession(){
   var fullname = $("#fullname").val();
   var dataString = 'fullname=' + fullname;

   $.ajax({
        type: "POST",
        url: "Page2.php",
        data: dataString,
        cache: false,
        success: function( data ) {
             alert("<?php echo $_SESSION["fullname"];?>");
            }
        });
      }
</script>

session_start();
$_SESSION["fullname"] = $_POST["fullname"];
exit();