Perturbations of giant magnons and single spikes in $\mathbb R \times S^2$

TL;DR

This paper investigates the stability of giant magnons and single spike solutions in a 2+1 dimensional spacetime, revealing that only zero modes are stable and providing numerical analysis for single spike perturbations.

Contribution

It introduces a simplified wave equation for perturbations of giant magnons and analyzes their stability, including numerical solutions for single spike perturbations.

Findings

Giant magnon perturbations reduce to a simple wave equation.

Only zero modes of giant magnons are stable under small deformations.

Numerical solutions indicate stability issues for single spike perturbations.

Abstract

Perturbations of giant magnons and single spikes in a $2+1$ dimensional $\mathbb R \times S^2$ background spacetime are analysed. Using the form of the giant magnon solution in the Jevicki-Jin gauge,the well-known Jacobi equation for small normal deformations of an embedded time-like surface are written down. Surprisingly, this equation reduces to a simple wave equation in a Minkowski background. The finiteness of perturbations and the ensuing stability of such giant magnons under small deformations are then discussed. It turns out that only the zero mode has finite deformations and is stable. Thereafter, we move on to explore the single spike solution in the Jevicki-Jin gauge. We obtain and solve the perturbation equation numerically and address stability issues.