Bill Troy - Open problems for the nonlinear Schrodinger equation

In 1972 the physicist Zakharov investigated solutions of the nonlinear cubic Schrodinger equation as a model of Langmuir plasma wave collapse. He assumed that solutions blow up in finite time in a self -similar, radially symmetric manner. For this  he derived an appropriate  complex, nonlinear, nonautonomous ODE with associated initial and boundary conditions. He found that physically relevant solutions must satisfy an integral condition representing  finite energy. For 51 years his problem has remained open. Here we discuss numerical and theoretical progress that has been made towards proving the existence of   finite energy solutions. We also discuss a simpler "real" problem that has remained open for over 50  years.