DCDS
Boundedness of solutions in a class of Duffing equations with a bounded restore force
Yiqian Wang
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 783-800 doi: 10.3934/dcds.2006.14.783
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
Communications on Pure & Applied Analysis 2009, 8(5): 1585-1606 doi: 10.3934/cpaa.2009.8.1585
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.

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