System of Kaup Equations with a Loaded Term in the Class of Periodic Functions

In this paper, a system of nonlinear Kaup equations with a loaded additional term in the class of periodic functions with respect to the spatial variable is considered. The invariance of the spectrum is proved and an analog of the Dubrovin system is derived for the evolution of the spectral parameters of a quadratic pencil of Sturm-Liouville operators on the entire line, the periodic coefficients of which are the solution of the Cauchy problem posed for the loaded system of nonlinear Kaup equations. Using trace formulas and an analog of the Dubrovin system, it is shown that the loaded system of nonlinear Kaup equations can be integrated by the inverse spectral problem method. An algorithm for solving the Cauchy problem for the loaded nonlinear Kaup’s system in the class of periodic functions with respect to the spatial variable is obtained. It is shown that if the initial functions are real analytic functions, then the solution will also be an analytic function with respect to the spatial variable. The 𝜋/2 -periodicity of the solution with respect to the spatial variable is revealed for the 𝜋/2 -periodicity of the initial functions.