Localization in One-dimensional Quasi-periodic Nonlinear Systems

To investigate localization in one-dimensional quasi-periodic nonlinear systems, we consider the Schrödinger equation $${\rm i}\dot{q}_n+\epsilon(q_{n+1}+q_{n-1})+V(n\tilde{\alpha}+x)q_n+ |q_n|^2q_n=0,\quad n\in\mathbb{Z},$$ as a model, with V a nonconstant real-analytic function on ${\mathbb{R}/\mathbb{Z}}$ , and ${\tilde{\alpha}}$ satisfying a certain Diophantine condition. It is shown that, if ${\epsilon}$ is sufficiently small, then for a.e. ${x\in\mathbb{R}/\mathbb{Z}}$ , dynamical localization is maintained for “typical” solutions in a quasi-periodic time-dependent way.