Collapse in the nonlinear Schrödinger equation of critical dimension $\{\mathbf{\sigma =1,\,D=2} \}$

Collapsing solutions to the nonlinear Schrödinger equation of critical dimension $\{\sigma =1,\,D=2 \}$ are analyzed in the adiabatic approximation. A three-parameter set of solutions is obtained for the scale factor $\lambda(t)$. It is shown that the Talanov solution lies on the separatrix between the regions of collapse and convenient expansion. A comparison with numerical solutions indicates that weakly collapsing solutions provide a good initial approximation to the collapse problem.