The unified transform method to the Hirota equation with periodic initial condition

Abstract

In this paper, we study the Hirota equation with periodic initial conditions via the unified transform method(UTM) extended by Fokas and Lenells. For nonlinear integrable evolution equations, the UTM expresses the solution in terms of a matrix Riemann-Hilbert problem. The associated jump matrices are computed in terms of the initial conditions and all boundary values. However, it is quite difficult to obtain all the initial conditions and boundary value data. It is known that the solution of the initial boundary value problem on a finite interval with $x$-periodic boundary conditions, can be regarded as the initial value problem on a circle. For the Hirota equation with periodic initial condition, we show that the solution can be explicitly expressed based on the initial data alone. Furthermore, the explicit solution is obtained, which corresponds to one-gap solution.