# Single and double linear and nonlinear flatband chains: spectra and modes

## TL;DR

This paper systematically analyzes flatband lattice modes in the diamond-chain model with nonlinear and linear on-site interactions, identifying various stationary states, localized modes, and their stability, including exact solutions and bifurcations.

## Contribution

It provides a comprehensive analysis of flatband modes in single and double diamond chains, including exact solutions, stability boundaries, and bifurcation phenomena, extending understanding of nonlinear flatband systems.

## Key findings

- Exact solutions for stationary states and flatbands
- Identification of stability regions for localized modes
- Discovery of symmetry-breaking bifurcations

## Abstract

We report results of systematic analysis of various modes in the flatband lattice, based on the diamond-chain model with the on-site cubic nonlinearity, and its double version with the linear on-site mixing between the two lattice fields. In the single-chain system, a full analysis is presented, first, for the single nonlinear cell, making it possible to find all stationary states, viz., antisymmetric, symmetric, and asymmetric ones, including an exactly investigated symmetry-breaking bifurcation of the subcritical type. In the nonlinear infinite single-component chain, compact localized states (CLSs) are found in an exact form too, as an extension of known compact eigenstates of the linear diamond chain. Their stability is studied by means of analytical and numerical methods, revealing a nontrivial stability boundary. In addition to the CLSs, various species of extended states and exponentially localized lattice solitons of symmetric and asymmetric types are studied too, by means of numerical calculations and variational approximation. As a result, existence and stability areas are identified for these modes. Finally, the linear version of the double diamond chain is solved in an exact form, producing two split flatbands in the system's spectrum.