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In dimension N=3 the cubic nonlinear Schrodinger equation has solutions whichbecome singular, i.e. at a spatial point they blow up to infinity in finitetime. In 1972 Zakharov famously investigated finite time singularity formationin the cubic nonlinear Schrodinger equation as a model for spatial collapse ofLangmuir waves in plasma, the most abundant form of observed matter in theuniverse. Zakharov assumed that (NLS) blow up of solutions is self-similar andradially symmetric, and that singularity formation can be modeled by a solutionof an associated self-similar, complex ordinary differential equation~(ODE). Aparameter a>0 appears in the ODE, and the dependent variable, Q, satisfies(Q(0),Q'(0))=(Q_{0},0), where Q(0)>0. A fundamentally important step towardsputting the Zakharov model on a firm mathematical footing is to prove, whenN=3, whether values a>0 and Q_{0}>0 exist such that Q also satisfies thephysically important `zero-energy' integral constraint. Since 1972 this hasremained an open problem. Here, we resolve this issue by proving that for everya>0 and Q(0)>0, Q satisfies the the `zero-energy' integral constraint.

Zero Energy Self-Similar Solutions Describing Singularity Formation In The Nonlinear Schrodinger Equation In Dimension N=3