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We use the inverse spectral problem method for integrating the nonlinear complex modified Korteweg-de Vries equation (cmKdV) with additional terms in the class of periodic infinite-gap functions. Also, we deduce the evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution to cmKdV. We prove that the Cauchy problem is solvable for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions. Moreover, we establish the solvability of the Cauchy problem for cmKdV with additional terms in the class of six times continuously differentiable periodic infinite-gap functions.
Хасанов et al. (Mon,) studied this question.