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.
Related
- Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: I. Bounded Perturbation
- Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity
- Lyapunov-Krasovsky Functionals and the Existence of Unbounded Solutions of Functional-Differential Equations of Retarded Type
- Aleksandrov-Bakelman-Pucci maximum principle for Ln-viscosity solutions of uniformly elliptic equations with unbounded terms: The critical case
- Fundamental solutions to Kolmogorov-Fokker-Planck equations with unbounded lower order terms
BibTeX
@misc{fu2026numerical,
title = {{Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: II. Unbounded Perturbation}},
author = {Mingwei Fu and Bin Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34194},
eprint = {2610.34194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34194},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: II. Unbounded Perturbation
AU - Mingwei Fu
AU - Bin Shi
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34194
UR - https://arxiv.org/abs/2610.34194
AB - 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.
ER -