On the dispersive shock waves of the defocusing Kundu–Eckhaus equation in an optical fiber

This paper develops the Whitham modulation theory to study dispersive shock waves (DSWs) of the defocusing Kundu–Eckhaus (KE) equation that models the propagation of the ultra-short femtosecond pulses in an optical fiber. We firstly use the perturbation expansion method and finite-gap integration theory to derive physical variables type and Riemann invariants type Whitham equations of the defocusing KE equation respectively. Then, by employing these Whitham equations, the basic rarefaction waves and DSWs caused by step-like initial value conditions are obtained, which are the key elements constituting solutions to the Riemann problem of the defocusing KE equation. Furthermore, the influences of non-Kerr nonlinear and self-frequency shift effects in the defocusing KE equation on these waves are analyzed, which have not been studied before. These notable influences include that the effects change the amplitudes of rarefaction waves, as well as the directions and magnitudes of DSWs corresponding to the hydrodynamic velocity function of the defocusing KE equation. Ultimately, we give a complete classification of solutions to the Riemann problem, which are divided into six cases, each of which illustrates the wave pattern change affected by the effects. This work may provide a physical mechanism for the generations and evolutions of DSWs in the optical fiber.