# Dynamics and stabilization of bright soliton stripes in the hyperbolic-dispersion nonlinear Schr\"odinger equation

## TL;DR

This paper investigates the dynamics and stability of bright soliton stripes in a two-dimensional nonlinear Schrödinger equation with hyperbolic dispersion, analyzing instabilities and stabilization methods.

## Contribution

It introduces a variational approximation for the transverse dynamics and demonstrates stabilization of soliton stripes using a channel-shaped potential.

## Key findings

- The variational approximation effectively models transverse instabilities.
- Channel-shaped potentials can fully stabilize bright soliton stripes.
- The study characterizes snaking and necking instabilities in the system.

## Abstract

We consider the dynamics and stability of bright soliton stripes in the two-dimensional nonlinear Schr\"odinger equation with hyperbolic dispersion, under the action of transverse perturbations. We start by discussing a recently proposed adiabatic-invariant approximation for transverse instabilities and its limitations in the bright soliton case. We then focus on a variational approximation used to reduce the dynamics of the bright-soliton stripe to effective equations of motion for its transverse shift. The reduction allows us to address the stripe's snaking instability, which is inherently present in the system, and follow the ensuing spatiotemporal undulation dynamics. Further, introducing a channel-shaped potential, we show that the instabilities (not only flexural, but also those of the necking type) can be attenuated, up to the point of complete stabilization of the soliton stripe.