arXiv subjects

Chengbin Zhu

Publications and source records attributed to Chengbin Zhu.

5 recordsLinked to original sources

arXiv2026

Modulational spectrum of infinite-depth hydroelastic Stokes waves

We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin--Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|\mu|})$, whereas the Benjamin--Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin--Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|\mu|)$ long-wave scale to $\cO(\sqrt{|\mu|})$ shows that the infinite-depth problem is singular.

math.AP↗

arXiv2026

Benjamin-Feir spectrum of hydroelastic Stokes waves

We determine the complete Benjamin-Feir spectrum near the origin for small-amplitude hydroelastic Stokes waves of the two-dimensional finite-depth irrotational Euler equations with surface tension and elastic bending. For the non-resonant Stokes branch and away from an intrinsic characteristic-collision surface $\mathfrak D$, we resolve all four Bloch eigenvalues bifurcating from the origin in the long-wave Floquet regime. Exploiting the Hamiltonian and reversible structure of the problem, we reduce the linearized Bloch operator to the four-dimensional spectral subspace bifurcating from the generalized kernel at the origin and conjugate the resulting matrix to the direct sum of a Benjamin-Feir block and a long-wave block. The long-wave pair remains purely imaginary, whereas the Benjamin-Feir pair is governed by an explicit closed-form instability index $\operatorname{Ind}(\mathtt{h},\kappa,b)$: a positive index produces a local figure-eight spectral curve with nonzero real part, while a negative index implies that all four small eigenvalues remain purely imaginary. Together with the Wilton-type resonance loci and the characteristic-collision surface $\mathfrak D$, this index yields a three-parameter spectral-stability diagram in the depth $\mathtt{h}$, surface tension $\kappa$, and bending rigidity $b$. The diagram recovers the classical pure-gravity critical-depth limit and, on the zero-bending boundary, the gravity--capillary stability diagram. It also reveals a genuinely hydroelastic phenomenon: all Wilton-type resonances disappear whenever $b\geq 1/14$ or $\kappa\geq 1/2$. This provides the first complete rigorous characterization of the local Benjamin-Feir spectrum for a hydroelastic free-boundary problem.

math.AP↗

arXiv2025

Synchronization Relations in the Hybrid Kuramoto Flow: Equivalence and a High-Coherence Criterion

We study four synchronization notions for the all-to-all hybrid Kuramoto model containing both first- and second-order oscillators with heterogeneous inertias and damping coefficients. We prove that full phase locking, bounded phase locking, and frequency synchronization are equivalent for arbitrary hybrid ensembles, and that each of these properties implies convergence of the complex order parameter. Conversely, if an order-parameter synchronized trajectory has limiting order parameter $Z^*$ satisfying $|Z^*|>\max\left\{\frac{\omega_M}{\lambda},\,1-\frac{2}{N}\right\}$, then the trajectory is frequency synchronized and hence fully phase locked. The converse proof is entirely real-dynamical. The omega-limit set is internally chain transitive and, because the order parameter is constant on that set, the dynamics reduce there to a product of frozen scalar equations. A tilted-energy argument shows that every nonstationary scalar factor of an internally chain transitive frozen set must cover the whole phase circle. Evaluating the constant mean field at an antipodal phase then forces $|Z^*|\le 1-2/N$, a contradiction. For zero natural frequencies, convergence follows from the analytic periodic gradient structure with degenerate inertia.

math.DS↗

arXiv2025

Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincar\'e inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.

math.NA↗

arXiv2025

On the Equivalence of Synchronization Definitions in the Kuramoto Flow: A Unified Approach

We present a rigorous mathematical framework establishing the equivalence of four classical notions of synchronization full phase-locking, phase-locking, frequency synchronization, and order parameter synchronization in generalized Kuramoto models, via a non-perturbative, finite-dimensional analysis. Our approach avoids linearization, mean-field limits, and restrictions on initial conditions, relying instead on global phase-space geometry, periodic vector field structure, and compactness arguments based on contradiction. These results clarify the foundational role of the order parameter and provide a unified understanding of synchronization across a broad class of heterogeneous oscillator networks.

math.DS↗