From the Polyakov string action using a conformal gauge we construct a threespin giant magnon solution describing a long open string in AdS 5 × S 5 which rotates both in two angular directions of S 5 and in one angular direction of AdS 5 . Through the Virasoro constraints the string motion in AdS 5 takes an effect from the string configuration in S 5 . The dispersion relation of the soliton solution is obtained as a superposition of two bound states of magnons. We show that there is a correspondence between a special giant magnon in AdS 2 and the sinh-Gordon soliton.

By using the non-diagonal uniform gauge for the Nambu-Goto string action we derive a gauge-fixed Hamiltonian of a square-root form for the closed string in AdS 5 ×S 5 which is wound and rotating in an angular direction in S 5 . From the Nambu-Goto string action using a non-diagonal gauge we construct a solution describing a wound string which rotates in the same angular direction as the winding direction. The relation between energy and angular momentum of the string solution is characterized by the winding number and the bending number, and becomes linear in the large angular momentum limit. The small angular momentum limit is compared with the strong coupling limit of the gauge-fixed Hamiltonian. We analyze a wound string solution which is rotating in the different angular direction from the winding direction. March, 2005

In the string theory in AdS 4 × CP 3 we construct the giant magnon and spike solutions with two spins in two kinds of subspaces of R t × CP 3 and derive the dispersion relations for them. For the single giant magnon solution in one subspace we show that its dispersion relation is associated with that of the big one-spin giant magnon solution in the RP 2 subspace. For the single giant magnon solution in the other complementary subspace its dispersion relation is similar to that of the one-spin giant magnon solution living in the S 2 subspace but has one additional spin dependence.

We construct a spinning closed string solution in AdS 5 × S 5 which is folded in the radial direction and has two equal spins in AdS 5 and a spin in S 5 . The energy expression of the three-spin solution specified by the folding and winding numbers for the small S 5 spin shows a logarithmic behavior and a one-third power behavior of the large total AdS 5 spin, in the long string and in the short string located near the boundary of AdS 5 respectively. It exhibits the non-regular expansion in the 't Hooft coupling constant, while it takes the regular one when the S 5 spin becomes large.

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