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

Abstract

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 \(\pi \)-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 \( {\pi }/{n} \), \(n\in \mathbb {N}\), \(n\ge 2 \), is the period of the initial function, then the number \({\pi }/{n} \) is the period for solving the Cauchy problem with respect to the variable \(x\).