The soliton perturbation theory based on the Riemann-Hilbert spectral problem

The nonlinear evolution equations associated with the general N × N Zakharov-Shabat spectral problem are considered in the Riemann-Hilbert problem framework. Equations describing the time-dependence of the Riemann-Hilbert spectral data in the presence of perturbation are derived. A scheme for calculating corrections to soliton solutions is described. A generalization of the theory is considered.