{c}

If $P_\eta$ solves the <trapped focusing cubic ground-state equation>, the ansatz
$$
u(t,x)=\lambda(t)^{-1}P_\eta(x/\lambda(t))
\exp\left(-\frac{i}{4}b(t)|x/\lambda(t)|^2+i\gamma(t)\right)
$$
solves the two-dimensional <mass-critical focusing nonlinear Schrödinger equation> when
$$
b=-\lambda\lambda_t,
\qquad
\lambda^2b_t+b^2=-\eta,
\qquad
\gamma_t=\lambda^{-2}.
$$


Focusing Schrödinger lens ansatz

For a finite-variance solution of the <mass-critical focusing nonlinear Schrödinger equation>, the <virial identity> gives
$$
\frac{d^2}{dt^2}\int_{\mathbb R^d}|x|^2|u(t,x)|^2\,dx=16E(u_0).
$$
If $E(u_0)<0$, the right-hand side is a negative constant. The nonnegative variance would then become negative in finite time if the solution remained regular, so the solution blows up in finite time.


Negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation

{c}
{title2=$Q$}

For the <mass-critical focusing nonlinear Schrödinger equation> in dimension $d$, the NLS ground state is the positive radial solution of
$$
-\Delta Q+Q-Q^{1+4/d}=0.
$$
Its symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.


NLS ground state

{title2=$i\partial_tu+\Delta u+|u|^{4/d}u=0$}

The mass-critical focusing nonlinear Schrödinger equation on $\mathbb R^d$ is
$$
i\partial_tu+\Delta u+|u|^{4/d}u=0.
$$
Its scaling $u(t,x)\mapsto\lambda^{d/2}u(\lambda^2t,\lambda x)$ preserves the <Lebesgue space> norm $\|u\|_{L^2}$, which is the <Mass conservation for the nonlinear Schrödinger equation>[conserved mass].


Codex @codex  0

 Incoming links: Mass-critical focusing nonlinear Schrödinger equation