Nonlinear Localized States on a Pyrochlore Lattice

Abstract

In the present work we explore a prototypical three-dimensional (3d) lattice possessing a flat band in the form of a pyrochlore lattice in the context of a dispersive nonlinear dynamical model, namely the discrete nonlinear Schrödinger (DNLS) equation. We set up the corresponding steady state and dynamical problems and discuss the linear spectrum of the relevant model before delving into a more detailed analysis of the nonlinear equilibria of the system. For the latter, we analyze the more well-established -- at the DNLS level -- fundamental discrete soliton states, as well as vortex structures. For the fundamental solitary waves, we connect their existence and stability with how they approach the linear bands. In the vortex case, we identify their stability features for vortices of topological charge S=1 and S=2 with those of the honeycomb and triangular lattices. An arguably even more intriguing feature of the pyrochlore lattice concerns the compactly supported nonlinear eigenstates stemming from the flat band of the linear spectrum. These compact localized modes are found to possess oscillatory instabilities for a range of propagation constants in the focusing case, although they can be stable in the latter, while they are found to be subject to symmetry-breaking instabilities in the defocusing nonlinearity case. These results offer a glimpse at the nexus of topology, flat band systems and dispersive nonlinear lattices in three spatial dimensions and as such may be a starting point toward a deeper exploration of such an intriguing interplay.

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BibTeX RIS

BibTeX

@misc{ramos2026nonlinear,
  title = {{Nonlinear Localized States on a Pyrochlore Lattice}},
  author = {Faustino Palmero Ramos and Avadh Saxena and Panayotis G. Kevrekidis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15855},
  eprint = {2606.15855},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15855},
}

RIS

TY  - UNPB
TI  - Nonlinear Localized States on a Pyrochlore Lattice
AU  - Faustino Palmero Ramos
AU  - Avadh Saxena
AU  - Panayotis G. Kevrekidis
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2606.15855
UR  - https://arxiv.org/abs/2606.15855
AB  - In the present work we explore a prototypical three-dimensional (3d) lattice possessing a flat band in the form of a pyrochlore lattice in the context of a dispersive nonlinear dynamical model, namely the discrete nonlinear Schrödinger (DNLS) equation. We set up the corresponding steady state and dynamical problems and discuss the linear spectrum of the relevant model before delving into a more detailed analysis of the nonlinear equilibria of the system. For the latter, we analyze the more well-established -- at the DNLS level -- fundamental discrete soliton states, as well as vortex structures. For the fundamental solitary waves, we connect their existence and stability with how they approach the linear bands. In the vortex case, we identify their stability features for vortices of topological charge S=1 and S=2 with those of the honeycomb and triangular lattices. An arguably even more intriguing feature of the pyrochlore lattice concerns the compactly supported nonlinear eigenstates stemming from the flat band of the linear spectrum. These compact localized modes are found to possess oscillatory instabilities for a range of propagation constants in the focusing case, although they can be stable in the latter, while they are found to be subject to symmetry-breaking instabilities in the defocusing nonlinearity case. These results offer a glimpse at the nexus of topology, flat band systems and dispersive nonlinear lattices in three spatial dimensions and as such may be a starting point toward a deeper exploration of such an intriguing interplay.
ER  -

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