Integration of negative-order modified Korteweg-de Vries equation in a class of periodic functions

We study the negative-order modified Korteweg-de Vries equation and show that it can be integrated by the inverse spectral transform method. We determine the evolution of the spectral data for the Dirac operator with periodic potential associated with a solution of the negative-order modified Korteweg-de Vries equation. The obtained results allow applying the inverse spectral transform method for solving the negative-order modified Korteweg-de Vries equation in the class of periodic functions. Important corollaries are obtained concerning the analyticity and the period of a solution in spatial variable. We show that a function constructed using the Dubrovin-Trubowitz system and the first trace formula satisfies the negative-order modified Korteweg-de Vries equation. We prove the solvability of the Cauchy problem for the infinite Dubrovin-Trubowitz system of differential equations in the class of three-times continuously differentiable periodic functions.