Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: II. Unbounded Perturbation

Abstract

Many nonlinear partial differential equations (PDEs) arising in physical models involve unbounded perturbations, where nonlinear terms contain spatial derivatives. Such perturbations can be expressed as compositions of unbounded operators with bounded nonlinear maps. In this paper, we propose alternating numerical schemes for constructing quasi-periodic solutions to this class of PDEs, including the nonlinear acoustic wave (NLAW) equation and the Korteweg-de Vries (KdV) equation. To render the nonlinear perturbations bounded, we apply inverse differential operators as a preconditioning; this, however, also rescales the linear part by polynomial factors in the spatial mode $n$. Two main difficulties arise. The first is to control the measure of the resulting ``bad'' frequencies, since the bounds on the measures of the excluded frequency sets are enlarged by polynomial factors in $n$; we show that this measure nevertheless remains small. The second concerns resonances involving the perturbed eigenvalues of the linearized operator at large scales. Our key observation is that, when the dispersive frequencies grow polynomially in $n$ with degree greater than one, such resonances cannot occur if the magnitude of the temporal index lies below a threshold proportional to $|n|$ raised to one less than that degree. It therefore suffices to consider the complementary regime, in which the growth induced by the unbounded perturbation can be controlled in terms of the temporal index. At large scales, an additional complication arises: for a given spatial index $n$, all temporal indices in a box may be ``bad''. We show, however, that at each scale there is at most one such spatial index. Consequently, the resulting counting loss is only polynomial and can be absorbed into the exponential-logarithmic threshold, allowing us once again to retain control over the measures of the excluded frequency sets.