## Analytic smoothing effect for solutions to Schrödinger equations with nonlinearity of integral type.(English)Zbl 1095.35045

Summary: We study analytic smoothing effects for solutions to the Cauchy problem for the Schrödinger equation $i\partial_tu+\frac 12\Delta u=f(u),$ with interaction described by the integral of the intensity with respect to one direction in two space dimensions $\bigl(f(u)\bigr)(t,x,y)= \lambda\left(\int^x_{-\infty}\bigl| u(t,x',y)\bigr|^2dx'\right)u (t,x,y).$ The only assumption on the Cauchy data is a weight condition of exponential type and no regularity assumption is imposed.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 78A60 Lasers, masers, optical bistability, nonlinear optics 35B65 Smoothness and regularity of solutions to PDEs