An Inverse Problem for a Nonlinear Schrödinger Equation

The existence of a local (in time) solution of (1.1) has been established by Ginibre and Velo [4], Kato [5], and others. For positive constant q, (1.1) may be considered as a model equation for the propagation of an intense laser beam through a medium with Kerr nonlinearity. The square of the transverse width of the incoming laser beam is proportional to q. Extensive investigations on the formation of the singularities of (1.1), with positive constant q, have been made by Fibich [3], Landman et al. [6], Landman et al. [7], LeMesurier et al. [8, 9], and others. In this paper, we consider q as a control and we study the following: