We consider the global regularity problem for defocusing nonlinear Schrödinger
systems
on Galilean spacetime
,
where the field
is
vector-valued,
is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order
outside of the unit ball
for some exponent
,
and
is a smooth, compactly supported forcing term. This generalises the
scalar defocusing nonlinear Schrödinger (NLS) equation, in which
and
. It is
well known that in the energy-subcritical and energy-critical cases when
or
and
, one has
global existence of smooth solutions from arbitrary smooth compactly supported initial data
and forcing
term
,
at least in low dimensions. In this paper we study the supercritical case where
and
.
We show that in this case, there exists a smooth potential
for some sufficiently
large
, positive and
homogeneous of order
outside of the unit ball, and a smooth compactly supported choice of initial data
and forcing
term
for which the solution develops a finite time singularity. In fact the solution is
locally discretely self-similar with respect to parabolic rescaling of spacetime.
This demonstrates that one cannot hope to establish a global regularity
result for the scalar defocusing NLS unless one uses some special property of
that equation that is not shared by these defocusing nonlinear Schrödinger
systems.
As in a previous paper of the author (Anal. PDE9:8 (2016), 1999–2030) considering the analogous
problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and
energy densities of
, then
itself, and then finally
design the potential
in order to solve the required equation.
Keywords
discretely self-similar blowup, finite time blowup,
nonlinear Schrödinger equation
