Cauchy Problem for the Loaded Korteweg-de Vries Equation in the Class of Periodic Functions

The inverse spectral problem method is applied to finding a solution of the Cauchy problem for the loaded Korteweg-de Vries equation in the class of periodic infinite-gap functions. A simple algorithm for constructing a high-order Korteweg-de Vries equation with loaded terms and a derivation of an analog of Dubrovin's system of differential equations are proposed. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and the first trace formula actually satisfies the loaded nonlinear Korteweg-de Vries equation. In addition, we prove that if the initial function is a real p-periodic analytic function, then the solution of the Cauchy problem is a real analytic function in the variable x as well, and also that if the number p/n, n. N, n = 2, is the period of the initial function, then the number p/n is the period for solving the Cauchy problem with respect to the variable x.