Nonlinear Schrödinger equation ($i{\psi }_t+\Delta \psi +F(|\psi {|}^2)\psi =0$)

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

The lens transform converts free Schrödinger equation evolution into evolution with a harmonic trapping potential by rescaling space, changing time and adding a quadratic phase. With the convention $i{\partial }_{s}u=-\frac{1}{2}\Delta u$ on ${\mathbb{R}}^d$, putThen $i{\partial }_{t}v=-\frac{1}{2}\Delta v+\frac{1}{2}|x{|}^{2}v$. The prefactor preserves the L2 space norm under the spatial change of variables. Infinite free time $s\to \pm \infty$ is represented by finite endpoints $t\to \pm \pi /2$.^[ref]

For $i{\partial }_{s}u=-\frac{1}{2}\Delta u+\lambda |u{|}^{p-1}u$, the transformed nonlinear coefficient is $\lambda (\cos t{)}^{d(p-1)/2-2}$. It remains constant precisely at $p=1+4/d$. This calculation explains why the transform is particularly useful for the mass-critical nonlinear Schrödinger equation. Its time interval and normalization are essential parts of the formula.