Integration of the Negative-Order Modified Korteweg–de Vries Equation with a Loaded Term in the Class of Periodic Functions

Spectral data of the Dirac operator with periodic potential are found. This operator isassociated with the negative-order modified Korteweg–de Vries equation with a loaded term. Theresults obtained permit one to construct a solution of the negative-order modified Korteweg–deVries equation with a loaded term in the class of periodic functions using the inverse spectralproblem method. The solvability of the Cauchy problem for an infinite system ofDubrovin–Trubowitz differential equations in the class of three times continuously differentiableperiodic functions is proved.