I will present a blow up construction - jointly with Kihyun Kim -
on the self-dual Chern-Simons-Schrödinger equation (CSS), also
known as a gauged nonlinear Schrödinger equation (NLS). CSS is
$L^{2}$-critical, admits solitons, and has the psuedoconformal symmetry.
These features are similar to the $L^{2}$-critical NLS. In this work,
we consider pseudoconformal blow-up solutions under $m$-equivariance,
$m\geq1$. Our result is threefold. Firstly, we construct a pseudoconformal
blow-up solution $u$ with given asymptotic profile $z^{\ast}$:
\[
\Big[u(t,r)-\frac{1}{|t|}Q\Big(\frac{r}{|t|}\Big)e^{-i\frac{r^{2}}{4|t|}}\Big]
e^{im\theta}\to z^{\ast}\qquad\text{in }H^{1}
\]
as $t\to0^{-}$, where $Q(r)e^{im\theta}$ 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 $u$. We construct a continuous family of solutions $u^{(\eta)}$,
$0\leq\eta\ll1$, such that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$
is a global scattering solution. Moreover, we exhibit a rotational
instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an abrupt spatial
rotation by the angle
\[
\Big(\frac{m+1}{m}\Big)\pi
\]
on the time interval $|t|\lesssim\eta$.

We are inspired by works in the $L^{2}$-critical NLS. In the seminal
work of Bourgain and Wang (1997), they constructed such pseudoconformal
blow-up solutions. Merle, Raphaël, and Szeftel (2013) showed an
instability of Bourgain-Wang solutions. Although CSS shares many features
with NLS, there are essential differences and obstacles over NLS.
Firstly, the soliton profile to CSS shows a slow polynomial decay
$r^{-(m+2)}$. This causes many technical issues for small $m$. Secondly,
due to the nonlocal nonlinearities, there are strong long-range interactions
even between functions in far different scales. This leads to a nontrivial
correction of our blow-up ansatz. Lastly, the instability
mechanism of CSS is completely different from that of NLS. Here, the phase rotation
is the main source of the instability. On the other hand, the self-dual
structure of CSS is our sponsor to overcome these obstacles. We exploited
the self-duality in many places such as the linearization, spectral
properties, and construction of modified profiles.

In the talks, I will present background of the problem, main theorems,
and outline of the proof with emphasis on heuristics of main features,
such as the long- range interaction between blow up profile and asymptotic
profile $z$, and rotational instability mechanism.