Article
Mathematics, Applied
Yingte Sun
Summary: This paper considers a class of nonlinear beam equations and proves the existence of many quasi-periodic solutions with non-resonant frequencies.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Yuan Zhang, Wen Si, Jianguo Si
Summary: In this paper, we consider quasi-periodically forced perturbations of dissipative Boussinesq systems with an elliptic fixed point in two cases: Hamiltonian case and reversible case. We prove the existence and linear stability of quasi-periodic solutions for the system with periodic boundary conditions. The method of proof is based on a Nash-Moser iterative scheme in the scale of Sobolev spaces developed by Berti and Bolle, but we have to adapt it substantially to deal with the specific system considered here.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Engineering, Mechanical
Xiufang Ren, Shiji Zhao
Summary: By modifying the coefficients of a one-dimensional quasi-periodic linear Schrodinger equation, the existence of quasi-periodic solutions is obtained and the properties of the reduced system are analyzed. The original result is based on infinite-dimensional KAM theory and bifurcation theory.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
M. Berti, L. Franzoi, A. Maspero
Summary: This study presents the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. Specifically, small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity are proven to exist, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all values of depth, gravity, and vorticity, and constrain the surface tension to a Borel set of asymptotically full Lebesgue measure.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics, Applied
Livia Corsi, Riccardo Montalto, Michela Procesi
Summary: In this study, almost-periodic solutions for quasi-linear perturbations of the Airy equation are proven to exist, marking the first result of its kind. These solutions are shown to be analytic in both time and space, achieved through a novel approach combining Craig-Wayne, KAM reducibility scheme, and pseudo-differential calculus on T-infinity.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Shengqing Hu
Summary: In this paper, we investigate the existence of quasi-periodic solutions for the quasi-periodic forced Schrodinger equation subject to Dirichlet boundary condition. Our approach is based on Birkhoff normal form theory and the KAM iterative method for infinite dimensional Hamiltonian systems.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Chuanfang Ge, Jiansheng Geng, Yingfei Yi
Summary: In this paper, we investigate the presence and stability of quasi-periodic breathers in granular chains of coupled Duffing oscillators with Hertzian interaction potential that has finite smoothness. By employing the Jackson-Moser-Zehnder analytic approximation technique and KAM iterations, we not only establish the existence and linear stability of quasi-periodic breathers, but also provide explicit estimates on the localization rate, etc., depending on the smoothness order and the number of oscillating frequencies.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Review
Mathematics
Lufang Mi, Jing Li
Summary: This paper discusses the existence of many quasi-periodic solutions of the cubic nonlinear Schrodinger equation with given potential V (x) using an infinite dimensional KAM theorem dealing with multiple eigenvalues.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
W-M Wang
Summary: This paper presents a set of smooth infinite energy global solutions to the non-integrable, nonlinear Schrodinger equations on R. These solutions are space-time quasi-periodic with two frequencies each.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics, Applied
Shengqing Hu, Jing Zhang
Summary: This paper investigates almost periodically forced harmonic oscillators and shows the existence of almost periodic solutions with the same frequencies as the forcing term in a positive Lebesgue measure set. The research extends previous results with quasi-periodic forcing terms and utilizes the Kolmogorov-Arnold-Moser theory.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
Ao Cai, Xueyin Wang
Summary: In this study, it is proven that for quasi-periodic Schrodinger operators in the local perturbative regime, with Diophantine frequency and sufficiently small C-k potential, the length of the spectral gap decays polynomially with respect to its label. The homogeneity of the spectrum is also demonstrated as an application of this result.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Weiping Yan, Binlin Zhang
Summary: This article considers the motion of relativistic strings in the Minkowski space R1+n, proving that they can exhibit a more generalized time quasi-periodic motion. These solutions are also timelike solutions.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Gerard Farre, Bassam Fayad
Summary: This article demonstrates the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori, addressing various stability theory problems including the existence of topologically unstable tori, optimality of exponential stability for Diophantine tori, and the existence of integrable Hamiltonians with orbits accumulating at infinity.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2022)
Article
Physics, Mathematical
Chuanfang Ge, Jiansheng Geng, Yingfei Yi
Summary: In this paper, we study the parameterized Newton's cradle lattice with Hertzian interactions and show the existence of small amplitude, linearly stable, quasi-periodic breathers. These breathers have 2b + 1 frequencies in time and are localized in space with a rate of 1/ |n|(1+alpha).
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Physics, Mathematical
Jiawen Luo, Zhenguo Liang, Zhiyan Zhao
Summary: This article discusses the behavior of a one-dimensional quantum harmonic oscillator perturbed by a linear operator with coefficients quasi-periodically depending on time. By establishing reducibility results, the growth of Sobolev norms, particularly the polynomial growth of H-s-norm, is observed when the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)