Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions

In this paper, the inverse spectral problem method is used to integrate the nonlinear mKdV–L equation in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of $6$ times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the system of Dubrovin equations and by using the first trace formula satisfies the mKdV–L equations. Moreover, we prove that if the initial function is a real-valued $\pi$-periodic analytic function, then the solution of the Cauchy problem for the mKdV–L equation is a real-valued analytic function in the variable $x$ as well; and if the number $\frac{\pi}{2}$ is a period (respectively, antiperiod) of the initial function, then the number $\frac{\pi}{2}$ is the period (respectively, antiperiod) in the variable $x$ of the solution of the Cauchy problem for the mKdV–L equations.