Integration of the Nonlinear Complex Modified Korteweg–de Vries (CMKdV) Equation with a Self-Consistent Source in the Form of a Series

Abstract To integrate the nonlinear complex modified Korteweg–de Vries (CMKdV) equation with a self-consistent source in the form of a series in the class of infinite-gap functions periodic with respect to the spatial variable, we apply the method of inverse spectral problem. We derive an infinite system of Dubrovin differential equations describing the evolution of the spectral data of a periodic Dirac operator whose coefficient is a solution of the CMKdV equation with a self-consistent source. We prove the solvability of the Cauchy problem for this system in the class of continuously differentiable periodic infinite-gap functions up to the sixth order inclusively. An algorithm for finding infinite-gap solutions of the Cauchy problem for the CMKdV equation with a self-consistent source is proposed, and an exact solution is found in the case of a single-gap Dirac operator.

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