Korteweg–de Vries

Intuition

Shallow water has a rule: taller waves move faster. A hump therefore steepens at its front, like a wave about to break on a beach. But shallow water also has dispersion, which spreads a hump out. The Korteweg–de Vries equation is the simplest model that contains both effects, and it has an exact hump shape where they cancel perfectly.

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Notice the shape never changes and the speed is constant. Taller solitons are narrower and faster. That is not a coincidence. Height, width and speed are locked together by one parameter.

Now watch two of them meet.

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The fast one catches the slow one, they merge into a single hump for a moment, and then both reappear exactly as before. The only trace of the collision is that the fast soliton is a little ahead of where it would have been, and the slow one is a little behind.

What if you start with a shape that is not a soliton?

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The bump sorts itself into a line of solitons, tallest first, leaving behind a small ripple that spreads away. The number and heights of the solitons that come out are determined by the initial shape alone. That is the inverse scattering transform at work.

The mathematics

The equation, in the normalisation this engine uses, is

$$u_t + 6\,u\,u_x + u_{xxx} = 0 .$$

The term $6 u u_x$ is the nonlinear steepening and $u_{xxx}$ is the dispersion. Look for a travelling wave $u(x,t) = f(x - ct)$ and you find the single-soliton solution

$$u(x,t) = \frac{c}{2}\,\operatorname{sech}^2\!\left(\frac{\sqrt{c}}{2}\,(x - ct - x_0)\right).$$

One parameter, the speed $c$, fixes the amplitude $c/2$ and the width $2/\sqrt{c}$.

The equation has infinitely many conserved quantities. The engine tracks the first three:

Name	Expression
Mass	$\int u \, dx$
Momentum	$\int u^2 \, dx$
Energy	$\int \left( u^3 - \tfrac{1}{2} u_x^2 \right) dx$

The energy is the Hamiltonian that generates the flow, $u_t = -\partial_x\, \delta H / \delta u$.

The numerics

On a periodic domain the engine writes the equation as $u_t = L u + N(u)$ with

$$L(k) = i k^3, \qquad N(u) = -3\,(u^2)_x .$$

In Fourier space $L$ is just a number for each wavenumber, so the linear part is handled exactly. The nonlinear part is computed in physical space and transformed back. This is the pseudo-spectral method. Time stepping uses ETDRK4, which treats the stiff dispersive term exactly and the nonlinear term to fourth order.

Validation

Test	Setup	Result
Single soliton	$c = 4$, $N = 512$, $L = 60$, $\Delta t = 10^{-3}$, $t = 5$	max error $5 \times 10^{-8}$
Two solitons	$c = 4$ and $c = 1$, 8000 steps	invariant drift below $10^{-8}$
Collision	$c = 6$ and $c = 1.5$	fast soliton amplitude restored to $3.0$ within $0.02$

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