Integration of a Nonlinear Korteweg–de Vries Equation with a Loaded Term and a Source
А. Б. Хасанов, T. G. Khasanov
Abstract
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 function series constructed with the use of the system of Dubrovin equations and the first trace formula indeed satisfies the loaded nonlinear Korteweg–de Vries equation with a source. In addition, it has been proved that if the initial function is a $$ \pi $$ -periodic real-analytic function, then the solution of the Cauchy problem is a real-analytic function with respect to the variable $$ x $$ as well; and if the number $$ \pi /n $$ is the period of the initial function, then the number $$ \pi /n $$ is the period of the solution of the Cauchy problem with respect to the variable $$ x $$ . Here $$ n\geqslant 2 $$ is a positive integer.