We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions, where the quadratic nonlinearity has the form N(∇u,∇v) = ∑k,l=1,2λkl (∂ku)(∂lv) with λ ε C. We prove that if the initial data u0 ε H6 ∩ H3,3 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states. The proof depends on the energy type estimates, smoothing property by Doi, and method of normal forms by Shatah.
- Global existence
- Nonlinear Schrödinger equations
- Quadratic nonlinearities
- Two spatial dimensions