Modified wave operators for nonlinear Schrödinger equations in lower order Sobolev spaces (Q2891098)

The authors study nonlinear Schrödinger equations of the form NEWLINE\[NEWLINE i u_t + \frac 12 \Delta u = \lambda_1 u^3 + \lambda_2 \overline u^2 u + \lambda_3 \overline u^3 + \lambda_0 |u|^{2 } u, \quad \text{on}\,\, {\mathbb R}\times{\mathbb R}, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE i u_t + \frac 12 \Delta u = \lambda_1 u^2 + \lambda_2 \overline u^2 + \lambda_0 |u| u , \quad \text{on}\,\, {\mathbb R}\times{\mathbb R}^2. NEWLINE\]NEWLINE For \(u_+\) small in suitable low order Sobolev spaces, the authors prove existence of a solution with final state \(u^+\). The arguments involve the construction of an approximate solution to the problem and a fixed point argument.