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 \[ iu_t-u_{xx}+T_vu+μ|u|^4u=0,\qquad x\in\mathbb T:=\mathbb R/(2π\mathbb Z). \] For every $κ\geq1$, 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 \[ \operatorname{dist}\left(\sum_j\ell_jλ_j,\mathbb Z\right) \geqγ_κ\prod_j(1+\ell_j^2\langle j\rangle^{2κ})^{-1} \] for some $γ_κ>0$ and every nonzero finitely supported integer sequence $\ell$, where $\langle j\rangle=\max\{1,|j|\}$. For this multiplier, every $3<M<4κ+\frac12$, every $A>0$, and all sufficiently small $μA^2>0$, the closed relative action tube of radius $3/10$ around $I_j=A\langle j\rangle^{-M}$ contains no full-dimensional invariant torus in $H^1(\mathbb T)$ 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.