SPATIOTEMPORAL DYNAMICS IN A TWISTED, CIRCULAR WAVEGUIDE ARRAY

The paper derives, to leading nontrivial orders in small coupling k, the single-core–dominated pulse with a0=ψ+k^2ã0+O(k^3), an=k^nãn, and θn=nφ+k^{N−2n}θ̃n, together with the recursive laws ã1=(ω−∂t^2)^{-1}ψ, ãn=(ω−∂t^2)^{-1}ã(n−1), and ã(N/2)=2cos(Nφ/2)(ω−∂t^2)^{-1}ã(N/2−1) (hence opposite-core suppression at φ=π/N), exactly as in the model’s construction . The solvability step L(ψ)ã0=−2ã1 with ker L(ψ)=span{ψ̇} is also the same (Fredholm alternative with even forcing) . For stability, the paper shows σess(L(c))=σess(L(0)) by Weyl’s theorem and computes the essential spectrum explicitly via Fourier decomposition, yielding σess=i(−∞,−α]∪i[α,∞) with α(φ)=minm(ω−2k cos(φ−2πm/N)), given in piecewise form and maximized at φ=π/N; this matches the model’s derivation and formula . Minor differences are cosmetic (e.g., the paper later reports sharper remainder orders supported by numerics), not substantive. Overall, the model reproduces the paper’s argument and outcomes.