Solitons of general topological charge over noncommutative tori

TL;DR

This paper explores solitons with arbitrary topological charge on noncommutative tori using time-frequency analysis, demonstrating that Gaussian-based projections minimize an energy functional and extend to the Moyal plane.

Contribution

It introduces a Gabor frame approach to describe solitons of general topological charge on noncommutative tori, including the Moyal plane case.

Findings

Gaussians are solitons of arbitrary topological charge on noncommutative tori.

Projections generated by Gaussians minimize the energy functional.

Gaussians are the only solitons on the Moyal plane.

Abstract

We continue the study of solitons over noncommutative tori from the perspective of time-frequency analysis and treat the case of a general topological charge. Solutions are associated with vector bundles of higher rank over noncommutative tori. We express these vector bundles in terms of vector-valued Gabor frames and apply the duality theory of Gabor analysis to show that Gaussians are solitons of general topological charge over noncommutative tori. An energy functional for projections over noncommutative tori is the basis for the self and anti-self duality equations of the solitons which turns out to have a reformulation in terms of Gabor atoms and we prove that projections generated by Gaussians minimize this energy functional. Finally, we comment on the case of the Moyal plane and the associated continuous vector-valued Gabor frames and show that Gaussians are the only class of solitons.