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

Ozawa, T.; Yamauchi, K.; Yamazaki, Y.

Osaka J. Math. 42, No. 4, 737-750 (2005).

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

Keywords:

pseudoconformal transformations; Stichartz estimates; nonlinear interaction; propagation of laser beams; Davey-Stewartson system