The Korteweg–de Vries (KdV) equation
\[
u_t + 6u\,u_x + u_{xxx}=0
\]
is one of the classic nonlinear wave equations. It combines two competing effects: the nonlinear term \(6u\,u_x\), which tends to steepen the wave, and the dispersive term \(u_{xxx}\), which tends to spread it out. Remarkably, these two effects can balance exactly and produce a solitary wave that travels without changing shape.
One exact solution is the one-soliton solution
\[
u(x,t)=2\kappa^2\,\mathrm{sech}^2\!\big(\kappa(x-4\kappa^2 t-x_0)\big),
\]
where \(\kappa>0\) controls the shape and speed, and \(x_0\) sets the initial center. This is called a soliton because it behaves like a localized pulse that keeps its form during propagation.
Several important properties follow directly from this formula. The peak amplitude is
\[
A=2\kappa^2,
\]
and the propagation speed is
\[
c=4\kappa^2.
\]
The characteristic width is inversely proportional to \(\kappa\):
\[
\text{width scale} \sim \frac{1}{\kappa}.
\]
Therefore, taller solitons are also narrower and faster. This is a distinctive feature of KdV solitons.
For the sample input \(\kappa=1\), the amplitude is
\[
A=2(1)^2=2,
\]
and the speed is
\[
c=4(1)^2=4.
\]
If the time range includes \(t=5\), then the center has moved to
\[
x_c(5)=x_0+ct=x_0+20.
\]
If \(x_0=0\), the soliton center is at \(x=20\) by that time.
What makes the KdV soliton especially important is that it is an exact nonlinear traveling wave, not just a small perturbation or approximate pulse. In many linear dispersive systems, a wave packet broadens as it travels. In contrast, the KdV soliton preserves its profile because the nonlinear steepening and dispersive spreading cancel each other exactly.
This balance is one of the central ideas in nonlinear wave theory. Solitons appear in many areas of physics and applied mathematics, including shallow-water waves, plasmas, internal waves, optical systems, and lattice dynamics. Although the full KdV equation can be solved numerically using methods such as pseudospectral schemes, the one-soliton solution gives an exact benchmark and an ideal way to understand the physics clearly.
At more advanced level, the KdV equation supports multiple solitons, and even after collisions these pulses re-emerge with their identities preserved, apart from phase shifts. That remarkable behavior is one reason soliton theory became so important in mathematical physics.
This calculator focuses on the exact one-soliton solution. It computes amplitude, speed, center position, and width scale, and it visualizes the stable propagation of the solitary wave over time.
