This appears to do what you need.

#!/usr/bin/perl -w
use strict;
while (my $righteq = <DATA>) {
    my @remaining = $righteq =~ /([a-z]\^(?:[a-z]|\([a-z][+-]\d+(?:\.\
+d+)?\)))/g;
    print '$remaining[',$_,'] is ',$remaining[$_],$/ for 0..$#remainin
+g;
}

__DATA__
+++-+++t^n+++f^n+e^(t-123.22)
t^n+--++-+f^n++e^(t-123.22)++
+++++---+t^n+f^n+++e^(t-123.22)-
[download]

Output:

$remaining[0] is t^n
$remaining[1] is f^n
$remaining[2] is e^(t-123.22)
$remaining[0] is t^n
$remaining[1] is f^n
$remaining[2] is e^(t-123.22)
$remaining[0] is t^n
$remaining[1] is f^n
$remaining[2] is e^(t-123.22)
[download]

Update: changed to read into $righteq instead of using $_.