Remarks on the nonlinear Schrödinger equations in plasma (Q1387459)

We consider the nonlinear Schrödinger equation: \[ i\Delta \varphi_t+ \Delta^2 \varphi +\beta \nabla \bigl(| \nabla \varphi |^{2p} \nabla\varphi \bigr)=0, \quad x \in \mathbb{R}^2,\;t>0, \tag{1} \] \(\varphi|_{t=0} =\varphi_0 (x)\), where \(p>0\), \(\beta \in\mathbb{R}\) are constants, \(\varphi_0 \in H^3 (\mathbb{R}^2)\). We notice the similarity between the Navier-Stokes equations and eq. (1). After choosing some special initial data, we show that for \(p\geq 1\), the solution blows up in finite time.