Nonpersistence of full-dimensional tori with polynomial decay in nonlinear Schrödinger equations
Abstract
We establish local nonpersistence of full-dimensional invariant tori with polynomially decaying actions for the periodic quintic nonlinear Schrödinger equation iut−uxx+Tvu+μ|u|4u=0, x∈𝕋:=ℝ/(2πℤ). For every κ≥1, we construct a fixed bounded real Fourier multiplier Tveijx=vjeijx whose frequencies λj=j2+vj satisfy the Bourgain-type Diophantine condition dist(∑jℓjλj,ℤ) ≥γκ∏j(1+ℓj2⟨ j⟩2κ)−1 for some γκ>0 and every nonzero finitely supported integer sequence ℓ, where ⟨ j⟩=max{1,|j|}. For this multiplier, every 3<M<4κ+1/2, every A>0, and all sufficiently small μA2>0, the closed relative action tube of radius 3/10 around Ij=A⟨ j⟩−M contains no full-dimensional invariant torus in H1(𝕋) carrying a complete flow, with arbitrary frequencies allowed. This advances the study of the polynomial-decay problem identified by Kuksin as a main open problem in KAM theory for one-dimensional Hamiltonian PDEs [Amer. Math. Soc. Transl. Ser. 2, 212 (2004), 237--258, Problem 7.1]. Bourgain discussed a related question for short-range Hamiltonian systems [Russian Math. Surveys 59 (2004), 231--246, Section 3]. The proof combines simultaneous Diophantine and weighted homological estimates, an exact common-phase transformation, and one normal-form step on four adjacent Fourier coordinates. Uniform phase rotation forces an action excursion, while overlapping parameter intervals cover all sufficiently small positive perturbations. The mechanism suggests applications to other Hamiltonian PDEs.
Related
- Scattering Across the Long-Range Threshold for One-Dimensional Nonlinear Schrödinger Equations
- Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
- Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space
- Polynomial mixing for the 3D damped cubic nonlinear Schrödinger equation with degenerate noise
- Power-Type Nonlinear Schrödinger Equations with the Logarithmic Laplacian
BibTeX
@misc{tong2026nonpersistence,
title = {{Nonpersistence of full-dimensional tori with polynomial decay in nonlinear Schrödinger equations}},
author = {Zhicheng Tong and Yong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22072},
eprint = {2610.22072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22072},
}
RIS
TY - UNPB
TI - Nonpersistence of full-dimensional tori with polynomial decay in nonlinear Schrödinger equations
AU - Zhicheng Tong
AU - Yong Li
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22072
UR - https://arxiv.org/abs/2610.22072
AB - We establish local nonpersistence of full-dimensional invariant tori with polynomially decaying actions for the periodic quintic nonlinear Schrödinger equation iu_t−u_xx+T_vu+μ|u|^4u=0, x∈𝕋:=ℝ/(2πℤ). For every κ≥1, we construct a fixed bounded real Fourier multiplier T_ve^ijx=v_je^ijx whose frequencies λ_j=j^2+v_j satisfy the Bourgain-type Diophantine condition dist(∑_jℓ_jλ_j,ℤ) ≥γ_κ∏_j(1+ℓ_j^2⟨ j⟩^2κ)^−1 for some γ_κ>0 and every nonzero finitely supported integer sequence ℓ, where ⟨ j⟩=max{1,|j|}. For this multiplier, every 3<M<4κ+1/2, every A>0, and all sufficiently small μA^2>0, the closed relative action tube of radius 3/10 around I_j=A⟨ j⟩^−M contains no full-dimensional invariant torus in H^1(𝕋) carrying a complete flow, with arbitrary frequencies allowed. This advances the study of the polynomial-decay problem identified by Kuksin as a main open problem in KAM theory for one-dimensional Hamiltonian PDEs [Amer. Math. Soc. Transl. Ser. 2, 212 (2004), 237--258, Problem 7.1]. Bourgain discussed a related question for short-range Hamiltonian systems [Russian Math. Surveys 59 (2004), 231--246, Section 3]. The proof combines simultaneous Diophantine and weighted homological estimates, an exact common-phase transformation, and one normal-form step on four adjacent Fourier coordinates. Uniform phase rotation forces an action excursion, while overlapping parameter intervals cover all sufficiently small positive perturbations. The mechanism suggests applications to other Hamiltonian PDEs.
ER -