Integration of equations of Kaup system kind with self-consistent source in class of periodic functions

In this paper, we consider the equations of Kaup system kind with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the considered nonlinear system of equations, which is based on the transformation to the spectral data of an associated quadratic pencil of Sturm–Liouville equations with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of equations of Kaup system kind with self-consistent source in the class of periodic functions. Moreover, it is shown that spectrum of the quadratic pencil of Sturm–Liouville equations with periodic coefficients associated with considering nonlinear system does not depend on time. In a one-gap case, we write the explicit formulae for solutions of the problem under consideration expressed in terms of the Jacobi elliptic functions. We show that if $p_{0} (x)$ and $q_{0} (x)$ are real analytical functions, the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients $p(x,t)$ and $q(x,t)$ are same. This implies that the solutions of considered problem $p(x,t)$ and $q(x,t)$ are real analytical functions in $x$.