Abstract
<title>Abstract</title> <p>In this study, the Schwarzian Korteweg–de Vries (sKdV) equation, a characteristic member of the theory of integrable systems, is investigated under Möbius periodic boundary conditions in order to model soliton dynamics in topologically twisted atomic lattice structures such as carbon nanorings, silicon chains and boron nitride. The spatial dimension is discretized by an operational matrix method based on q-Legendre polynomials evaluated at q-Gauss–Lobatto collocation points; the differentiation matrix is constructed via barycentric Lagrange interpolation and is verified to converge, in the q→1 limit, to the classical Legendre–Gauss–Lobatto spectral derivative at machine precision (of order 10⁻¹³). During a symbolic verification carried out in this work, the rational nonlinear structure of the sKdV equation is re-examined and it is confirmed that, under the transformation w = u_x, the equation reduces to the standard potential mKdV form (w_t = w_xxx − 6w²w_x). Unlike the v = √u_x transformation proposed in the original draft, this form contains no rational singularity and can be integrated stably and directly using the explicit fourth-order Runge–Kutta (RK4) scheme. The Möbius periodic boundary condition is integrated into the system by a direct substitution technique. Systematic numerical experiments reveal two distinct blow-up mechanisms: (i) a numerical (aliasing) blow-up that appears, even without a boundary condition, when the initial soliton is chosen too narrow relative to the resolution, and which is removable by increasing resolution; and (ii) a genuine finite-time singularity that arises when the anti-periodic Möbius condition (c=0, a=−1, d=1, b=0) is imposed even with a well-resolved initial condition (in the regime where free evolution is stable), and which is brought forward — rather than removed — as resolution increases. The second mechanism is shown to converge to a fixed T* value in the Δt→0 limit, confirming that the observation is a genuine finite-time blow-up rather than a numerical instability. For four parameter sets labelled by representative atomic bond types (C–C, Si–Si, B–N, C–O), the lattice deformation coefficient (q) and the boundary barrier parameter (c) are swept; both the dimensionless blow-up time T* and the early-time (measured at T*/2) energy/amplitude conservation ratios are found to vary markedly across the sweep. The (q,c) values are treated as illustrative parameter labels reflecting qualitative bond-stiffness and electronegativity trends, not as first-principles-derived constants. The parameter set with q=0.99, c=0.00 (labelled Pure Carbon, C–C) yields the highest early-time stability score (96.77%), consistent with the protective effect expected of a stiff, low-barrier sp²-type lattice. The two stability criteria (blow-up time and early-time conservation) yield different orderings, demonstrating that the effect of the c parameter is non-monotonic. The physical origin of the boundary-induced blow-up is attributed to the amplification, by the nonlinear structure of the mKdV equation, of a small symmetry-breaking perturbation generated by the discrete imposition of the Möbius coupling. The method itself is validated against an exact analytic mKdV kink solution, reproducing it with spectral (exponential) accuracy. The primary contribution of this work is therefore methodological and diagnostic: a reusable q-Legendre spectral framework, together with a carefully characterized cautionary result on boundary-induced finite-time blow-up, rather than a claim of successful long-time soliton transport.</p>