Integration of the Modified Korteweg-de Vries Equation with Time-Dependent Coefficients and a Self-Consistent Source

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 problem, we find Lax pairs and employ the inverse scattering method. Note that in the case under study the Dirac operator is not selfadjoint, and so the eigenvalues can be multiples. We find the equations describing the dynamics in time of the scattering data of a nonselfadjoint Dirac operator whose potential is a solution to the modified Korteweg-de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions. As a special case, we examine a loaded modified Korteweg-de Vries equation with a self-consistent source. The equations describe the dynamics in time of the scattering data of a nonselfadjoint Dirac operator whose potential is a solution to the loaded modified Korteweg-de Vries equation with variable coefficients in the class of rapidly decreasing functions. We provide some examples that illustrate applications of the results.