On Pseudoconformal Blow-Up Solutions to the Self-Dual Chern-Simons-Schrödinger Equation: Existence, Uniqueness, and Instability

We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is L 2 L^{2} -critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the L 2 L^{2} -critical NLS. In this work, we consider pseudoconformal blow-up solutions under m m -equivariance, m ≥ 1 m\geq 1 . Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution u u with given asymptotic profile z ∗ z^{\ast } : \[ [ u ( t , r ) − 1 | t | Q ( r | t | ) e − i r 2 4 | t | ] e i m θ → z ∗ in  H 1 \Big [u(t,r)-\frac {1}{|t|}Q\Big (\frac {r}{|t|}\Big )e^{-i\frac {r^{2}}{4|t|}}\Big ]e^{im\theta }\to z^{\ast }\qquad \text {in }H^{1} \] as t → 0 − t\to 0^{-} , where Q ( r ) e i m θ Q(r)e^{im\theta } is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of u u . We construct a continuous family of solutions u ( η ) u^{(\eta )} , 0 ≤ η ≪ 1 0\leq \eta \ll 1 , such that u ( 0 ) =