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Zuhong You

Publications and source records attributed to Zuhong You.

7 recordsLinked to original sources
arXiv2026

KAM theory for quasi-periodic equilibrium of generalized Frenkel-Kontorova model in $\mathbb{Z}^{d}$

We extend the KAM theory for one dimensional Frenkel-Kontorova model to multidimensional case in quasi-periodic media. By using the quasi-Newton method, we reduce the linearized operator with quasi-periodic coefficient in the Nash-Moser iteration to the composition of two diagonal operators and a near-identity operator, such that the small divisor problem can be overcome by imposing the simultaneous Diophantine condition. In contrast, we also discuss the application of Craig-Wayne-Bourgain method to the quasi-periodic equilibrium problem.

math.DS↗
arXiv2026

Anderson localization for 1-d quasi-periodic Schrödinger operators with degenerate weights

We establish Anderson localization for 1-d discrete Schrödinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and quasi-periodic. Operators of this kind originate from distinct mathematical physics problems, which include the Frenkel-Kontorova model with impurities, the discretization of singular Sturm-Liouville operators, and the Fisher-KPP lattice equation in heterogeneous media.

math-ph↗
arXiv2026

A Parameterization of Small Amplitude KP Finite Gap Solutions via Classical Schottky Uniformization and Persistence of One and Two Gap Solutions

We develop a parameterization of small amplitude finite gap solutions to the KP equation with prescribed spatial wavenumber vectors using classical Schottky uniformization. This parameterization is suitable for a Lyapunov-Schmidt reduction. Using this parameterization, we prove the persistence, under Hamiltonian perturbations, of periodic one-gap solutions for both KP-I and KP-II and of bi-periodic two-gap solutions for KP-I.

math.AP↗
arXiv2026

Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schrödinger Equations via Resonant Mode Control

This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schrödinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set $\mathcal{K}$ to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts.

math.DS↗
arXiv2025

Construction of Quasi-periodic solutions with the same Gevrey index as nonlinear terms in Multi-Dimensional NLS

We investigate the persistency of quasi-periodic solutions to multi-dimensional nonlinear Schrödinger equations (NLS) involving Gevrey smooth nonlinearity with an arbitrary Gevrey index $α>1$. By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of quasi-periodic solutions that are Gevrey smooth with the same Gevrey index as the nonlinearity.

math.AP↗