I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,

$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^1( \mathbb{R} ^d ) $

adding a phase term to initial solution makes for a blowup in finite time, e.g. :

$\psi _0 = R^ {(0)} _B (r) $, the solitonic solution, exsists globaly, but it is said that adding a phase term such as $\psi _0 = R^ {(0)} _B (r) e^ {i r^2} $, would suffice for a blowup at some $0 < Z_c < \infty $.

There is some motivation for this in the "Lense Transformation" of the Nonlinear Schrodinger (NLS), but I wasn't able to find any proof, numerics or further intuition for why it should work in the BNLS.

Any thoughts?