Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation

We study the linear stability analysis and nonlinear excitations on a continuous wave background for the fifth-order nonlinear Schrödinger equation. We find that there are a modulation stability (MS) quasi-elliptic ring and a MS curve in the modulation instability (MI) regime. The concrete expressions of different types of nonlinear waves are obtained by using Darboux transformation. And we present the relations and phase diagram between MI and these waves. We discover that the antidark (AD) soliton and nonrational W-shaped soliton can only exist on the resonance line of outside of the MS quasi-elliptic ring and the right of the MS curve. We perform numerical simulation to test the stability of the AD soliton. By introducing the perturbation energy, we further show that solitons in the MI regime are caused by both the fourth- and fifth-order effects.