Small solutions to nonlinear Schrödinger equations
A local existence theorem is proved for solutions of the Cauchy problem for the nonlinear Schrödinger equation of the following form: \[ iu_ t=-\Delta u+iP(u,\nabla u,\overline u,\nabla \overline u), \] where \(u\) is a complex function of time and \(n\) spatial variables, and \(P\) is a polynomial function (without constant or linear terms) of a degree \(s\). The existence of solutions at finite values of time is proved, using assumptions which establish boundedness of the initial data. The main technical ingredient in proving the theorem is the so-called smoothing effect of Kato type. The theorem is proved separately for four different cases corresponding, respectively, to \(n=1\) and \(n \geq 2\), and \(s=2\) and \(s \geq 3\). In each case, different specific boundedness conditions should be imposed on the initial data in order to prove the local existence of the solution.
scientific article; zbMATH DE number 438585
scientific article