Effective dynamics and defect expansions for polynomial PDEs on thin annuli

TL;DR

This paper develops a geometric and analytic framework for polynomial PDEs on thin annuli, demonstrating dimension reduction to effective 1D dynamics and introducing Sobolev orthogonal polynomial methods for multiscale analysis.

Contribution

It introduces a unified approach combining Sobolev orthogonal polynomials, dimension reduction, and homogenization for polynomial PDEs on thin geometries, applicable to integrable and non-integrable models.

Findings

Solutions converge to effective 1D dynamics on the limiting circle.

Transverse defect correctors describe anisotropic dispersive effects.

The framework is robust under geometric and polynomial Hilbert space changes.

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 Schr\"odinger, 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.