Nonlinear Schrödinger

Intuition

A light pulse in a glass fibre is a fast oscillation wrapped in a slow envelope. The glass has dispersion, so different colours in the pulse travel at different speeds and the envelope spreads. But intense light also changes the glass’s refractive index, which bends the phase of the wave in proportion to its intensity. For the right pulse shape the two effects cancel and the envelope holds together. That is an optical soliton, and the same equation describes Bose–Einstein condensates and deep-water wave groups.

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The bright line is the envelope $|\psi|$. The faint line is the real part, the oscillation underneath. The envelope keeps its shape while the carrier runs through it.

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Two envelopes meet, interfere during the overlap, and separate intact.

Now start with twice the soliton amplitude.

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It does not travel. It breathes. This is the Satsuma–Yajima solution, two solitons bound together at one point, oscillating with period $\pi/2$.

The mathematics

The focusing cubic nonlinear Schrödinger equation is

$$i\,\psi_t + \tfrac{1}{2}\,\psi_{xx} + |\psi|^2 \psi = 0 .$$

Here $\psi$ is complex. Its magnitude is the envelope and its phase carries the oscillation. The bright soliton is

$$\psi(x,t) = a\,\operatorname{sech}\big(a (x - v t - x_0)\big)\, \exp\!\Big( i\,v x + \tfrac{i}{2}\,(a^2 - v^2)\,t \Big).$$

Unlike KdV, amplitude $a$ and velocity $v$ are independent. Galilean invariance lets you boost any soliton to any speed.

Conserved quantities tracked by the engine:

Name	Expression
Mass	$\int \lvert\psi\rvert^2 \, dx$
Momentum	$\operatorname{Im}\int \psi^* \psi_x \, dx$
Energy	$\int \left( \tfrac{1}{2}\lvert\psi_x\rvert^2 - \tfrac{1}{2}\lvert\psi\rvert^4 \right) dx$

The numerics

Written as $\psi_t = L\psi + N(\psi)$:

$$L(k) = -\tfrac{i}{2}\,k^2, \qquad N(\psi) = i\,|\psi|^2 \psi .$$

Same machinery as KdV: Fourier pseudo-spectral in space, ETDRK4 in time.

Validation

Test	Setup	Result
Moving bright soliton	$a = 1$, $v = 1$, $N = 512$, $L = 40\pi$, $\Delta t = 10^{-3}$, $t = 10$	max error $3 \times 10^{-9}$
Two solitons	10000 steps	invariant drift below $10^{-8}$

Source: crates/soliton-core/src/equations/nls.rs and its tests.