Integration of the modified Korteweg–de Vries equation with time-dependent coefficients and with a self-consistent source

In this paper, we consider the Cauchy problem for the modified Korteweg–de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions. To solve the stated problem, the inverse scattering method is used. Lax pairs are found, which will make it possible to apply the inverse scattering method to solve the stated Cauchy problem. Note that in the case under consideration the Dirac operator is not self-adjoint, so the eigenvalues can be multiple. Equations are found for the dynamics of change in time of the scattering data of a non-self-adjoint operator of the Dirac operator with a potential that is a solution of the modified Korteweg–de Vries equation with variable time-dependent coefficients and with a self-consistent source in the class of rapidly decreasing functions. A special case of a modified Korteweg–de Vries equation with time-dependent variable coefficients and a self-consistent source, namely, a loaded modified Korteweg–de Vries equation with a self-consistent source, is considered. Equations are found for the dynamics of change in time of the scattering data of a non-self-adjoint operator of the Dirac operator with a potential that is a solution of the loaded modified Korteweg–de Vries equation with variable coefficients in the class of rapidly decreasing functions. Examples are given to illustrate the application of the obtained results.