Integration of a non-linear Hirota type equation with additional terms

In this paper, the inverse spectral problem method is used to integrate a Hirota type equation with additional terms 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 six times continuously differentiable periodic infinite-gap functions is proved. It is also shown that the Cauchy problem is solvable at all times for sufficiently smooth initial conditions.