The Cauchy problem for a nonlinear Hirota equation in the class of periodic infinite-zone functions

The method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. The evolution of spectral data is introduced for the periodic Dirac operator whose coefficient is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is real-analytic and $\pi$-periodic, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function of the variable $x$; next, if the number $\pi/2$ is a period (antiperiod) of the initial function, then the number $\pi/2$ is a period (antiperiod) in the variable $x$ for the solution of the Cauchy problem for the Hirota equation.