A SYSTEM OF NONLINEAR SCHRÖDINGER EQUATIONS WITH DELTA-FUNCTIONS AS INITIAL DATA (Qualitative Theory on ODEs and their applications to Mathematical Modeling)

Abstract

This paper treats a system of nonlinear Schrödinger equations with delta-functions as initial data. By imposing delta-fUnctions on the initial data, the partial differential equations are reduced into a couple of ODEs, and the behaviors of the solutions are observed in detail. Doi-Shimizu [2] considered a similar problem in case that the powers of nonlinearities coincides in both equations. But this paper removes the coincidence of the powers of nonlinearities, classifies the decay estimates of the global solutions in cases of dissipative nonlinearities, and proves the existence of blowing-up solution in cases that both nonlinearities are amplification.