Abstract
In this paper, the method of the inverse spectral problems are used to integrate the nonlinear modified Korteweg-de Vries equation with finite density in the class of periodic functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five-fold continuously differentiable periodic functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies the mKdV equations with finite density.
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