Integration of a Nonlinear Sine-Gordon–Liouville-Type Equation in the Class of Periodic Infinite-Gap Functions

The method of inverse spectral problem is used to integrate a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions. The evolution of the spectral data for the periodic Dirac operator is introduced in which the coefficient of the Dirac operator is a solution of the nonlinear sine-Gordon–Liouville-type equation. The solvability of the Cauchy problems is proved for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin differential equations and the first-trace formula satisfies the sine-Gordon–Liouville-type equation.