A numerical study of DNLS dynamics on quasiperiodic tight-binding chains — finite-size scaling, long-time evolution, and the self-trapping threshold gap. To our knowledge, the first numerical study of DNLS dynamics on a tribonacci substitution chain. 📦
We integrate the discrete nonlinear Schrödinger (DNLS) equation on two quasiperiodic chains — the Fibonacci chain (n=2) and the Rauzy–tribonacci chain (n=3) — starting from mid-gap eigenstates of the linear Hamiltonian, over nonlinearity strengths λ ∈ [0, 10] and times T ∈ [50, 10⁶], tracking the inverse participation ratio (IPR) as the localization measure.
The tribonacci chain exhibits differential nonlinear robustness: at canonical parameters (T=50, N=500, λ=1.5) the tribonacci mid-gap state retains >95% of its linear IPR while the Fibonacci state loses ~57%. The mechanism is the stronger multifractal spatial hierarchy of the Rauzy–tribonacci eigenstate.
Tribonacci vs Fibonacci IPR retention at canonical parameters.
Tribonacci mid-gap state is more localized before nonlinearity acts (0.0820 vs 0.0210).
Perron–Frobenius eigenvalue of the tribonacci companion matrix; unique real root of x³ − x² − x − 1 = 0 in [1,2].
First numerical study of DNLS dynamics on a tribonacci substitution chain, to our knowledge.