# Ultrashort dark solitons interactions and nonlinear tunneling in the modified nonlinear Schr\"odinger equation with variable coefficients

## TL;DR

This paper investigates the dynamics, interactions, and tunneling behaviors of ultrashort dark solitons in an inhomogeneous fiber modeled by a variable coefficient modified nonlinear Schrödinger equation, highlighting effects of self-steepening and barrier interactions.

## Contribution

It introduces a detailed analysis of dark soliton behavior in variable coefficient NLSE with self-steepening, including shock wave formation, stability under noise, and tunneling characteristics.

## Key findings

- Self-steepening causes shock wave formation in dark solitons.
- Dark solitons maintain shape after tunneling through barriers.
- Tunneling behavior depends on barrier height and soliton amplitude.

## Abstract

We present the study of the dark soliton dynamics in an inhomogenous fiber by means of a variable coefficient modified nonlinear Schr\"{o}dinger equation (Vc-MNLSE) with distributed dispersion, self-phase modulation, self-steepening and linear gain/loss. The ultrashort dark soliton pulse evolution and interaction is studied by using the Hirota bilinear (HB) method. In particular, we give much insight into the effect of self-steepening (SS) on the dark soliton dynamics. The study reveals a shock wave formation, as a major effect of SS. Numerically, we study the dark soliton propagation in the continuous wave background, and the stability of the soliton solution is tested in the presence of photon noise. The elastic collision behaviors of the dark solitons are discussed by the asymptotic analysis. On the other hand, considering the nonlinear tunneling of dark soliton through barrier/well, we find that the tunneling of the dark soliton depends on the height of the barrier and the amplitude of the soliton. The intensity of the tunneling soliton either forms a peak or valley and retains its shape after the tunneling. For the case of exponential background, the soliton tends to compress after tunneling through the barrier/well.