Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces (Q1607878)

Summary: We study the asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces, \[ i \partial_t u + ({1}/{2}) \Delta u = \mathcal{N}(u),\quad (t,x) \in \mathbb{R} \times \mathbb{R}^2; \qquad u(0,x) = \varphi(x), \quad x \in \mathbb{R}^2, \] where \[ \mathcal{N}(u) = \sum_{j,k=1}^{2}(\lambda_{jk} (\partial_{x_j}u)(\partial_{x_k}u) + \mu_{jk}(\partial_{x_j} \overline{u})(\partial_{x_k}\overline{u})), \] with \(\lambda_{jk},\mu_{jk} \in \mathbb{C}\). We prove that if the initial data \(\varphi\) satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has an asymptotic representation in the neighborhood of the scattering states.