Solutions of the generalized nonlinear Schrödinger equation in two spatial dimensions

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Abstract
Recently, from two independent methods, the generalized nonlinear Schrödinger evolution equation in two spatial dimensions has been derived both by Ablowitz and Haberman and by Morris. Here the same extension of the Schrödinger cubic equation is obtained from a two‐dimensional spatial inversionlike integral equation when a suitable time dependence is introduced. We investigate the solutions (corresponding to degenerate kernels of the inversion equation) in both cases, where the nonlinear part reduces or not to a cubic term. While in the first case, the solutions are not confined, on the contrary in the second case, we show explicitly for any finite time, that there exists an infinite number of solutions which are confined in the two‐dimensional coordinate space.