| Abstract: |
| In dimension $N=3$ the cubic nonlinear Schrodinger (NLS) equation has solutions which become singular, i.e.
at a spatial point they blow up to infinity in finite time.
In 1972 Zakharov famously investigated finite time singularity formation in
% introduced zero-energy self-similar solutions of
the cubic nonlinear Schrodinger equation as a model for
spatial collapse
of Langmuir waves in plasma, the most abundant form of observed matter in the universe.
Zakharov assumed that (NLS) blow up of solutions
is self-similar and radially symmetric, and that
singularity formation
can be modeled by a solution of an associated self-similar, complex ordinary differential equation~(ODE).
A parameter $a>0$ 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 towards putting the Zakharov model on a firm mathematical footing is to
prove, when $N=3,$
whether values $a>0$ and $Q_{0}>0$ exist such that $Q$ also satisfies the
physically important `zero-energy` integral constraint.
Here, we discuss this issue and present several open problems. |
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