Defocusing cubic nonlinear Schrödinger equation in two dimensions ($i{u}_t+\Delta u-|u{|}^{2}u=0$)

The two-dimensional defocusing cubic nonlinear Schrödinger equation is mass-critical: the scaling $u(t,x)\mapsto {\lambda }^{-1}u(t/{\lambda }^2,x/\lambda )$ preserves the $L^2$ norm. The defocusing sign makes its nonlinear energy positive.

The harmonic-oscillator energy space isThe spatial moment prevents mass from escaping to infinity.

The embedding $\Sigma \hookrightarrow L^2({\mathbb{R}}^d)$ is compact. Rellich compactness controls a fixed ball, whilecontrols the tail uniformly.

Critical points of the trapped defocusing cubic energysatisfy $-\Delta v+|x{|}^{2}v-\omega v+v^3=0$.

A concrete Schrödinger expanding lens ansatz for the mass-critical equation starts with a finite-energy solution $v\in \Sigma$ of the trapped defocusing stationary equationSet $\lambda (t)=\sqrt{1+t^2}$. Thensolves the Defocusing cubic nonlinear Schrödinger equation in two dimensions. Direct substitution gives the scale/chirp conditions $b=-t=-\lambda {\lambda }_t$, ${\lambda }^2{b}_t+b^2=-1$, and phase rate ${\gamma }_t=-\mu /{\lambda }^2$. The displayed stationary equation supplies exactly the remaining spatial terms.^[ref]

The harmonic-oscillator energy space assumption controls both the ordinary gradient and the quadratic-phase gradient. Moreover,by the substitution $x=\lambda y$ and ${\int }_{\mathbb{R}}(1+t^2{)}^{-1}dt=\pi$. This gives an explicit free solution to which scattering from a finite Strichartz norm applies.

In two dimensions, the exponents $(q,r)=(4,4)$ give the Strichartz estimate for the free Schrödinger equation below, since $2/q+2/r=1$. The homogeneous estimate and its dual areThe second operator is the adjoint of the first, so the bound follows by Lp duality. It also defines the time integral by continuous extension when an ordinary absolutely convergent integral is unavailable.

The pair $(4,4)$ is the critical cubic spacetime pair. The excluded two-dimensional endpoint is $(2,\infty )$, so the displayed bound should not be called that endpoint estimate.^[ref]

If a global cubic Schrödinger solution has finite $L_{t,x}^4$ norm on a time ray, the dual Strichartz estimate makes its interaction representation Cauchy in $L^2$. It therefore converges to scattering data $u_{\infty }$, and $u(t)-e^{it\Delta }{u}_{\infty }\to 0$ in $L^2$.