# Effects of the third-order dispersion on continuous waves in complex potentials

## TL;DR

This paper investigates how third-order dispersion influences continuous wave solutions in complex potentials within nonlinear Schrödinger equations, revealing stabilization effects and conditions for weak stability in optical waveguides.

## Contribution

It introduces a method to construct complex potentials supporting specific phase-gradient continuous waves and analyzes the stabilizing role of third-order dispersion on modulational instability.

## Key findings

- Third-order dispersion suppresses modulational instability.
- Weak stability occurs when MI growth rate is small.
- Zero state stability is also examined.

## Abstract

A class of constant-amplitude (CA) solutions of the nonlinear Schrodinger equation with the third-order spatial dispersion (TOD) and complex potentials are considered. The system can be implemented in specially designed planar nonlinear optical waveguides carrying a distribution of local gain and loss elements, in a combination with a photonic-crystal structure. The complex potential is built as a solution of the inverse problem, which predicts the potential supporting a required phase-gradient structure of the CA state. It is shown that the diffraction of truncated CA states with a correct phase structure can be strongly suppressed. The main subject of the analysis is the modulational instability (MI) of the CA states. The results show that the TOD term tends to attenuate the MI. In particular, simulations demonstrate a phenomenon of weak stability, which occurs when the linear-stability analysis predicts small values of the MI growth rate. The stability of the zero state, which is a nontrivial issue in the framework of the present model, is studied too