Integration of the nonlinear Korteweg—de Vries equation with loaded term and source

A simple algorithm for deriving an analog of the system of Dubrovin differential equations is proposed. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula really satisfies the loaded nonlinear Korteweg—de Vries equation with a source. In addition, it has been proven that if the initial function is a real $\pi$-periodic analytic function, then the solution of the Cauchy problem is also a real analytic function with respect to the variable $x$; and if the number $\pi/n$ 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 $x$. Here $n$ is a natural number, $n\geqslant 2$.