DCDS
Boundedness of solutions in a class of Duffing equations with a bounded restore force
Yiqian Wang
In this paper, we consider the boundedness of all the solutions and the existence of quasi-periodic solutions for Duffing equations

$\frac{d^2x}{dt^2}+\arctan x=p(t),$

where $p(t+1)=p(t)$ is a smooth function.

keywords: KAM theorem. Boundedness of solutions bounded restore force
CPAA
The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation
Lei Jiao Yiqian Wang
In this paper, we construct small amplitude quasi-periodic solutions for one dimensional nonlinear Schrödinger equation

i$u_t=u_{x x}-mu-f(\beta t,x)|u|^2 u,$

with the boundary conditions

$u(t,0)=u(t,a\pi)=0, \ -\infty < t < \infty,$

where $m$ is real and $f(\beta t,x)$ is real analytic and quasi-periodic on $t$ satisfying the non-degeneracy condition

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(\beta t,x)dt\equiv f_0=$ const., $\quad 0\ne f_0 \in\mathbb R,$

with $\beta\in\mathbb R^b$ a fixed Diophantine vector.

keywords: Schrödinger equation KAM Hamiltonian systems normal form.
DCDS
A new method for the boundedness of semilinear Duffing equations at resonance
Zhiguo Wang Yiqian Wang Daxiong Piao
We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.
keywords: periodic nonlinearity Moser's theorem. boundedness Hamiltonian system at resonance

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