Finite-size effects for single spike

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-Δ'' relation for both R _t × S ² and R _t × S ³ cases.