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 integrable within the framework of the AKNS scheme is reviewed. The approach is a modification of the known finite-band integration method, based on the re-parametrization of the solution with the use of algebraic resolvent of the polynomial defining the solution in the finite-band integration method. This approach permits one to obtain periodic solutions in an effective form necessary for applications. The periodic solutions are found for such systems as the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation, the Heisenberg model, the uniaxial ferromagnet, the AB system, and self-induced transparency and stimulated Raman scattering equations. The modulation Whitham theory 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 the nonlinear theory of modulational instability describing the transformation of a local disturbance expanding into a nonuniform region presented as a modulated periodic wave whose evolution is governed by the Whitham equations. This theory explains the formation of solitons on the sharp front of a long pulse.