The Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions

In this paper, the method of the inverse spectral problem is applied to finding a solution to the Cauchy problem for the Korteweg-de Vries equation in the class of periodic infinite-gap functions. A simple derivation of the system of differential Dubrovin equations is proposed. The solvability of the Cauchy problem is proved for the infinite system of Dubrovin differential equations in the class of four-times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly convergent functional series constructed using the solution of the infinite system of Dubrovin equations and the formula for the first trace, does satisfy the nonlinear Korteweg–de Vries equation. In addition, it is proved that if the number $\frac{\pi }{n}$ is the period of the initial function, then the number $\frac{\pi }{n}$ is the period for the solution of the Cauchy problem with respect to the variable $x$. Here $n\ge 2$ is a positive integer.