The Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with a Loaded Term
Abstract
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 $$\frac {\pi }{2} $$ is the period (or antiperiod) of the initial function then $$\frac {\pi }{2} $$ is the period (antiperiod) of the solution of the Cauchy problem problem with respect to $$x$$ .