Effective dynamics and defect expansions for polynomial PDEs on thin annuli
Jean-Pierre Magnot
TL;DR
This work develops a constructive geometric framework for polynomial PDEs on thin planar annuli by introducing renormalized Sobolev norms and Sobolev orthogonal polynomials, which yield stable Galerkin approximations tailored to the thin geometry. A general dimension-reduction theorem shows that solutions converge to an effective one-dimensional dynamics on the circle $S^1$, with transverse defect correctors captured by cell problems that describe anisotropic dispersion and homogenization effects. The authors demonstrate robustness of the reduction and numerical schemes under changes of Sobolev order and oscillatory coefficients, and apply the theory to purely tangential integrable models (KdV, mKdV, NLS, sine–Gordon) and anisotropic dispersive systems (Zakharov–Kuznetsov), as well as to dissipative, forced, and homogenized settings. The Sobolev-orthogonal polynomial approach provides a constructive, multiscale toolkit linking dimension reduction, homogenization, and (asymptotic) integrability in thin geometries, with potential computational benefits for PDE analysis in singular domains.
Abstract
We develop a geometric and analytic framework for polynomial partial differential equations posed on thin annuli in the plane. Using renormalized Sobolev inner products, we construct Sobolev orthogonal polynomial bases adapted to the thin geometry and use them to define stable Galerkin approximations. We prove a general dimension-reduction theorem for polynomial Hamiltonian and dissipative PDEs, showing that solutions converge to effective one-dimensional dynamics on the limiting circle. Beyond the leading-order limit, we identify transverse defect correctors and derive cell problems describing anisotropic dispersive and homogenized effects. Our framework applies uniformly to integrable models (KdV, modified KdV, nonlinear Schrödinger, sine--Gordon), anisotropic dispersive systems such as Zakharov--Kuznetsov, and non-integrable perturbations including dissipation, forcing, and rapidly oscillating coefficients. We establish stability of the effective dynamics under changes of Sobolev order and of polynomial Hilbert geometry, and show robustness of the associated Galerkin schemes. The results provide a unified geometric perspective on dimension reduction, homogenization, and integrability in thin geometries, and introduce Sobolev orthogonal polynomial methods as a constructive tool for multiscale PDE analysis.