Existence and emergent dynamics of quadratically separable states to the Lohe tensor model

TL;DR

This paper explores quadratically separable states in the Lohe tensor model, revealing their existence and dynamics, and extends understanding of tensor aggregation models by connecting them to double matrix models.

Contribution

It introduces and analyzes quadratically separable states in the Lohe tensor model, expanding the class of solutions beyond rank-1 tensors and linking to double matrix models.

Findings

Quadratically separable states exist in the Lohe tensor model.

Emergent dynamics of these states can be studied via the double matrix model.

The approach generalizes previous solutions like rank-1 tensors.

Abstract

A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously well-studied aggregation models on the space of low-rank tensors such as the Kuramoto model, Lohe sphere and matrix models as special cases. Due to its structural complexities in cubic interactions for the Lohe tensor model, explicit construction of solutions with specific structures looks daunting. Recently, we obtained completely separable states by associating rank-1 tensors. In this paper, we further investigate another type of solutions, namely "{\it quadratically separable states}" consisting of tensor products of matrices and their component rank-2 tensors are solutions to the double matrix model whose emergent dynamics can be studied using the same methodology of the Lohe matrix model.