The Cauchy Problem for a Nonlinear Hirota-Type Equationwith a Self-Consistent Series Source

In this paper the inverse spectral problem method is applied to integrate a nonlinear Hirota-type equation with a self-consistent series source in the class of periodic infinite-gap functions. An infinite system of differential equations is derived that describes the evolution of spectral data for the periodic Dirac operator. The solvability of the Cauchy problem for this system in the class of six-times continuously differentiable periodic infinite-gap functions is proved. In addition, an algorithm for finding infinite-gap solution to the Cauchy problem for a Hirota-type equation with a self-consistent series source is proposed, and an exact solution is found in the case of the single-gap case for the Dirac operator.