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Spiky strings in Bethe Ansatz at strong coupling

We study spiky string solutions in AdS3 x S1 that are characterized by two spins S,J as well as winding m in S1 and spike number n. We construct explicitly two-cut solutions by using the SL(2) asymptotic Bethe Ansatz equations at leading order in strong coupling. Unlike the folded spinning string, these solutions have asymmetric distributions of Bethe roots. The solutions match the known spiky string classical results obtained directly from string theory for arbitrary semiclassical parameters, including J=0 and any value of S, namely short and long strings. At large spins and winding number the string touches the boundary, and we find a new scaling limit with the energy given as E-S = n /2/ pi sqrt[1+ 4 pi^2/n^2 (J^2/ln^2 S + m^2/ln^2 S)] ln S. This is a generalization of the known scaling for the folded spinning string.