Small Denominators and Subresonant Accumulation in Weakly Nonlinear Dispersive Dynamics

Abstract

We study a small-denominator mechanism in weakly nonlinear dispersive dynamics. After Fourier decomposition, a nonlinear dispersive equation becomes an infinite system of weakly coupled oscillators. Higher-order correction terms may then contain infinite families of nonresonant Fourier interactions whose detunings tend to zero. Such families do not produce exact secular terms, but their accumulated contribution may grow as a power of time. We call this effect subresonant accumulation. The rigorous part of the paper is the analysis of a model forced oscillator and of an abstract subresonant Duhamel sum. If the detuning and coefficients have the form $Δ_n\sim c n^{-p}$ and $B_n\sim b n^{-κ}$, then the accumulated contribution grows as $t^{1-α}$, where $α=(κ-1)/p$. We then show how this mechanism appears in a quartic Fourier family for the Klein--Gordon dispersion law. For the full nonlinear partial differential equation we formulate a conditional approximation result: provided that all remaining resonant and almost resonant interactions are controlled, the subresonant term gives the leading long-time correction.