Solvability of the Cauchy problem of non-isotropic Schrödinger equations in Sobolev spaces

We study the existence of solutions for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation iut+Δu+|u|αu+aux1x1x1x1+bux1x1x1x1x1x1=0, where a, b are not simultaneously vanishing real constants, α is a positive constant, and x=(x1,x2)∈R2. By using Kato's method, we establish some local existence results for initial data belonging to Hs(R2), where s≥0 if either b≠0, 0<α≤3, or b=0, a≠0, 0<α≤83, s≥1−3α if b≠0, α>3, and s≥1−38α if b=0, a≠0, α>83. Global existence is also established under some additional conditions.