Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on generic tori dimension d≥1. study the behavior high Sobolev norms Hs, s≫1, solutions initial conditions in Hs whose Hρ-Sobolev norm, 1≪ρ≪s, is smaller than ε≪1. provide control Hs-norm over time interval order O(ε−2). Due to lack conserved quantities controlling norms, key ingredient proof construction modified energy equivalent “low norm” Hρ (when ρ sufficiently high) nontrivial This achieved by means normal form techniques for quasi-linear involving para-differential calculus. The main difficulty possible loss derivatives due small divisors arising three waves interactions. By performing “tame” estimates we obtain upper bounds higher Hs.