Quasi-periodic Solutions for One-Dimensional Nonlinear Lattice Schrödinger Equation with Tangent Potential

In this paper, we construct time quasi-periodic solutions for the nonlinear lattice Schrödinger equation ${\rm i}\dot{q}_n+\epsilon (q_{n+1}+q_{n-1}) +\tan\pi(n\tilde{\alpha}+x)q_n+\epsilon|q_n|^2q_n=0$, $n\in\mathbb{Z},$ where $\tilde{\alpha}$ satisfies a certain Diophantine condition and $x\in\mathbb{R}/\mathbb{Z}$. We prove that for $\epsilon$ sufficiently small, the equation admits a family of small-amplitude time quasi-periodic solutions for “most” of $x$ belonging to $\mathbb{R}/\mathbb{Z}$.