We use the reduction of the string dynamics on AdS 4 × CP 3 to the Neumann-Rosochatius integrable system. All constraints can be expressed simply in terms of a few parameters. We analyze the giant magnon and single spike solutions on R t × CP 3 with two angular momenta in detail and find the energy-charge relations. The finite-size effects of the giant magnon and single spike solutions are analyzed.

We consider strings moving in the R t × S 3 η subspace of the η-deformed AdS 5 × S 5 and obtain a class of solutions depending on several parameters. They are characterized by the string energy and two angular momenta. Finite-size dyonic giant magnon belongs to this class of solutions. Further on, we restrict ourselves to the case of giant magnon with one nonzero angular momentum, and obtain the leading finite-size correction to the dispersion relation.

We use the reduction of the string dynamics on R t × S 3 to the Neumann-Rosochatius integrable system to map all string solutions described by this dynamical system onto solutions of the complex sine-Gordon integrable model. This mapping relates the parameters in the solutions on both sides of the correspondence. In the framework of this approach, we find finite-size string solutions, their images in the (complex) sine-Gordon system, and the leading finite-size effects of the single spike "E − ∆ϕ" relation for both R t × S 2 and R t × S 3 cases.Recent developments in AdS/CFT correspondence between type IIB strings on AdS 5 ×S 5 and its dual N = 4 super Yang-Mills (SYM) theories [1] are mainly based on the integrabilities discovered in both theories. Integrability of the SYM side appears in the calculations of conformal dimensions which are related to the string energies according to the AdS/CFT correspondence. A remarkable observation by Minahan and Zarembo [2] is that the conformal dimension of an operator composed of scalar fields in the N = 4 SYM in the planar limit can be computed by diagonalizing the Hamiltonian of one-dimensional integrable spin chain model. This task can be done by solving a set of coupled algebraic equations called Bethe ansatz equations. It has been shown that explicit calculation of the eigenvalues for various SYM operators agree with those computed from the SYM perturbation theory. This result has been further extended to the full P SU(2, 2|4) sector [3] and the Bethe ansatz equations which are supposed to hold for all loops are conjectured [4,5].The string side of the correspondence is mostly studied at the classical level due to the lack of full quantization. The type IIB string theory on AdS 5 × S 5 is described by a nonlinear sigma model with P SU(2, 2|4) symmetry [6]. This sigma model has been shown to have an infinite number of local and nonlocal conserved currents [7] and some of the conserved charges such as energy and angular momentum are computed explicitly from the classical integrability (see, for example, [8] and the references therein). These results based on the classical integrability provide valuable information on the AdS/CFT duality in the domain of large t'Hooft coupling constant. A new direction to quantizing the string theory is to find exact S-matrix between the fundamental spectrum of the theory on the world sheet. It has been shown that the S-matrix along with the exact particle spectrum can be determined by the underlying symmetry and integrability in the theory [9,10]. The overall scalar factor of the S-matrix, sometimes called as dressing phase, which can not be determined by the symmetry alone, has been computed exactly [11,12] from the crossing relation [13]. In this process, the explicit expression of the dressing phase in the classical limit, which was determined by the classical integrability, was essential [14].Various classical solutions play an important role in testing and understanding the correspondence. The classical giant magnon (GM) state [15] ...