Whitham modulation theory and dam-breaking problem under periodic solutions to the defocusing Hirota equation

Abstract

We explore the Whitham modulation theory and one of its physical applications, the dam-breaking problem for the defocusing Hirota equation that describes the propagation of ultrashort pulses in optical fibers with third-order dispersion and self-steepening higher-order effects. By using the finite-gap integration approach, we deduce periodic solutions of the equation and discuss the degeneration of genus-one periodic solution to a soliton solution. Furthermore, the corresponding Whitham equations based on Riemann invariants are obtained, which can be used to modulate the periodic solutions with step-like initial data. These Whitham equations with the weak dispersion limit are quasilinear hyperbolic equations and elucidate the averaged dynamics of the fast oscillations referred to as dispersive shocks, which occur in the solution of the defocusing Hirota equation. We analyze the case where both characteristic velocities in genus-zero Whitham equations are equal to zero and the values of two Riemann invariants are taken as the critical case. Then by varying these two values as step-like initial data, we study the rarefaction wave and dispersive shock wave solutions of the Whitham equations. Under certain step-like initial data, the point where two genus-one dispersive shock waves begin to collide at a certain time, that is, the point where the genus-two dispersive shock wave appears, is investigated. We also discuss the dam-breaking problem as an important physical application of the Whitham modulation theory.