Single spike solutions for strings on S-2 and S-3

We study solutions for rigidly rotating strings on a two-sphere. Among them we find two limiting cases that have a particular interest, one is the already known giant magnon and the other we call the single spike solution. The limiting behavior of this last solution is a string infinitely wrapped around the equator. It differs from that solution by the existence of a single spike of height (theta) over bar that points toward the north pole. We study its properties and compute its energy E and angular momentum J as a function of (theta) over bar. We further generalize the solution by adding one angular momentum to obtain a solution on S-3. We find a spin chain interpretation of these results in terms of free fermions and the Hubbard model but the exact relation with the same models derived from the field theory is not clear