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Spiky strings in AdS_3 x S^1 and their AdS-pp-wave limits

We study a class of classical solutions for closed strings moving in AdS_3 x S^1 part of AdS_5 x S^5 with energy E and spin S in AdS_3 and angular momentum J and winding m in S^1. They have rigid shape with n spikes in AdS_3. We find that when J or m are non-zero, the spikes do not end in cusps. We consider in detail a special large n limit in which S ~ n^2, J ~ n, i.e. S >> J >> 1, with (E+S)/ n^2, (E-S)/ n, J/n, m/n staying finite. In that limit the spiky spinning string approaches the boundary of AdS_5. We show that the corresponding solution can be interpreted as describing a periodic-spike string moving in AdS_3 --pp-wave x S^1 background. The resulting expression for the string energy should represent a strong-coupling prediction for anomalous dimension of a class of dual gauge theory states in a particular thermodynamic limit of the SL(2) spin chain.