On Pseudoconformal Blow-Up Solutions to the Self-Dual Chern-Simons-Schrödinger Equation: Existence, Uniqueness, and Instability
Authors: Kim Kihyun , Kwon Soonsik 
Published: April 2023

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 }

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket u left-parenthesis t comma r right-parenthesis minus StartFraction 1 Over StartAbsoluteValue t EndAbsoluteValue EndFraction upper Q left-parenthesis StartFraction r Over StartAbsoluteValue t EndAbsoluteValue EndFraction right-parenthesis e Superscript minus i StartFraction r squared Over 4 StartAbsoluteValue t EndAbsoluteValue EndFraction Baseline right-bracket e Superscript i m theta Baseline right-arrow z Superscript asterisk Baseline in upper H Superscript 1">

[

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
)

=
u

u^{(0)}=u

and for

η
>
0

\eta >0

,

u

(
η
)

u^{(\eta )}

is a global scattering solution. Moreover, we exhibit a rotational instability as

η
→

0

+

\eta \to 0^{+}

:

u

(
η
)

u^{(\eta )}

takes an abrupt spatial rotation by the angle
\[

(

m
+
1

m

)

π

\Big (\frac {m+1}{m}\Big )\pi

\]
on the time interval

|

t

|

≲
η

|t|\lesssim \eta

.

We are inspired by works in the

L

2

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
)

r^{-(m+2)}

. This causes many technical issues for small

m
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.