THE CAUCHY PROBLEM FOR THE LOADED KORTEWEG-DE VRIES EQUATION WITH A SOURCE IN THE CLASS OF PERIODIC INFINITE-GAP FUNCTIONS (Scientific journal of the Fergana State University, 2023)
In this paper, the inverse spectral problem method is used to integrate the loaded Korteweg-de Vries (KdV) equation with a source in the class of periodic infinite-gap functions. We introduce the evolution of the spectral data of the Hill operator whose coefficient is a solution of the loaded KdV equation with a source. The solvability of the Cauchy problem for an infinite system of Dubrovin's differential equations is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin's equations and the first trace formula indeed satisfies the loaded KdV equation with a source.