Dirac Solitons and Topological Edge States in the $\beta$-Fermi-Pasta-Ulam-Tsingou dimer lattice

TL;DR

This paper derives and analyzes Dirac solitons and topological edge states in a nonlinear dimer lattice modeled after the $eta$-FPUT system, revealing their properties, boundary behaviors, and stability through analytical and numerical methods.

Contribution

It introduces a continuum Dirac-type model for the nonlinear dimer lattice and characterizes the existence, boundary conditions, and stability of nonlinear edge states.

Findings

Derivation of Dirac soliton profiles and conservation laws.

Identification of boundary conditions supporting edge states.

Numerical verification of soliton stability and dynamics.

Abstract

We consider a dimer lattice of the Fermi-Pasta-Ulam-Tsingou (FPUT) type, where alternating linear couplings have a controllably small difference, and the cubic nonlinearity ($\beta$-FPUT) is the same for all interaction pairs. We use a weakly nonlinear formal reduction within the lattice bandgap to obtain a continuum, nonlinear Dirac-type system. We derive the Dirac soliton profiles and the model's conservation laws analytically. We then examine the cases of the semi-infinite and the finite domains and illustrate how the soliton solutions of the bulk problem can be ``glued'' to the boundaries for different types of boundary conditions. We thus explain the existence of various kinds of nonlinear edge states in the system, of which only one leads to the standard topological edge states observed in the linear limit. We finally examine the stability of bulk and edge states and verify them through direct numerical simulations, in which we observe a solitary wave setting into motion due to the instability.