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

We study the existence of solutions for the Cauchy problem of the non-isotropically perturbed nonlinear Schrödinger equation iut+Δu+|u|αu+aux1x1x1x1+bux1x1x1x1x1x1=0iut+Δu+|u|αu+aux1x1x1x1+bux1x1x1x1x1x1=0, where aa, bb are not simultaneously vanishing real constants, αα is a positive constant, and x=(x1,x2)∈R2x=(x1,x2)∈R2. By using Kato’s method, we establish some local existence results for initial data belonging to Hs(R2)Hs(R2), where s≥0s≥0 if either b≠0b≠0, 0<α≤30<α≤3, or b=0b=0, a≠0a≠0, View the MathML source0<α≤83, View the MathML sources≥1−3α if b≠0b≠0, α>3α>3, and View the MathML sources≥1−38α if b=0b=0, a≠0a≠0, View the MathML sourceα>83. Global existence is also established under some additional conditions.