Quasi-periodic Solutions for Nonlinear Schrodinger Equations with Legendre Potential

In this paper, the nonlinear Schrödinger equations with Legendre potential iut − uxx + VL (x)u + mu + secx · |u|²u = 0 subject to certain boundary conditions is considered, where V L ( x ) = − 1 2 − 1 4 tan 2 x , x ∈ (−π/2, π/2). It is proved that for each given positive constant m > 0, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.