# Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schr\"odinger equation

## TL;DR

This paper investigates the dynamics of solitons in a nonlocal, $ ext{PT}$-symmetric nonlinear Schr"odinger equation using variational methods, revealing behaviors like trapping, blow-up, and soliton interactions.

## Contribution

It develops a variational collective coordinate approach to analyze soliton dynamics in the Ablowitz-Musslimani nonlocal NLS, including collision and blow-up phenomena.

## Key findings

- Single soliton can pass, decay, or blow up at the origin.
- Variational approximation captures finite-time blow-up.
- Two solitons exhibit mass transfer and scattering.

## Abstract

We study the integrable nonlocal nonlinear Schr\"odinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.