The Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with a Loaded Term

The method of inverse spectral problems is applied for integrating the defocusing nonlinear Scrödinger (DNS) equation with loaded terms in the class of infinite-gap periodic functions. We describe the evolution of the spectral data for a periodic Dirac operator whose coefficient is a solution to the DNS equation with loaded terms. We prove the following assertions. (1) It the initial function is real-valued, -periodic, and analytic then the solution of the Cauchy problem for the DNS equation with loaded terms is a real-valued analytic function in x. (2) If 2 is the period (or antiperiod) of the initial function then 2 is the period (antiperiod) of the solution of the Cauchy problem problem with respect to x.