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, $\pi$-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 $\pi/2$ is the period (or antiperiod) of the initial function then $\pi/2$ is the period (antiperiod) of the solution of the Cauchy problem problem with respect to $x$.