NEW APPROACH TO PERIODIC-SOLUTIONS OF INTEGRABLE EQUATIONS AND NONLINEAR-THEORY OF MODULATIONAL INSTABILITY

A new method of finding the periodic solutions for the equations integ rable within the framework of the AKNS scheme is reviewed. The approac h is a modification of the known finite-band integration method, based on the re-parametrization of the solution with the use of algebraic r esolvent of the polynomial defining the solution in the finite-band in tegration method. This approach permits one to obtain periodic solutio ns in an effective form necessary for applications. The periodic solut ions are found for such systems as the nonlinear Schrodinger equation, the derivative nonlinear Schrodinger equation, the Heisenberg model, the uniaxial ferromagnet, the AB system, and self-induced transparency and stimulated Raman scattering equations. The modulation Whitham the ory describing the slow modulation of periodic waves is expressed in a form convenient for applications. The Whitham equations are obtained for all abovementioned cases. The technique developed is applied to th e nonlinear theory of modulational instability describing the transfor mation of a local disturbance expanding into a nonuniform region prese nted as a modulated periodic wave whose evolution is governed by the W hitham equations. This theory explains the formation of solitons on th e sharp front of a long pulse.