Normally, when two waves on water meet, they pass through each other and their heights add up. When they move on, they might change shape or scatter. But in some very specific conditions, waves can behave much more like solid objects.

These special waves are called solitons.

The Great Wave of Translation

The story of the soliton begins in 1834 when a Scottish engineer named John Scott Russell was observing a boat on a canal. When the boat suddenly stopped, it pushed forward a single, well-defined hump of water. Russell chased this wave on horseback for miles as it travelled at a steady speed without losing its shape. He called it a "Wave of Translation."

The Korteweg–de Vries Equation

It wasn't until 1895 that mathematicians Diederik Korteweg and Gustav de Vries derived an equation to explain this phenomenon. This equation, now known as the KdV equation, describes how waves travel in shallow water.

The KdV equation is written as:

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

Here, $u$ is the height of the wave, $t$ is time, and $x$ is position. This equation balances two opposing forces:

• Nonlinearity ($6u u_x$): This part tends to make the wave steeper. Taller parts of the wave move faster than shorter parts.
• Dispersion ($u_{xxx}$): This part makes the wave spread out as it travels.

When these two forces perfectly balance each other out, the wave can travel indefinitely without changing its shape—a solitary wave.

Particle-Like Collisions

The most amazing property of solitons is how they interact. If a taller, faster soliton catches up to a shorter, slower one, they don't just splash and dissipate. Instead, they merge briefly and then separate, emerging with their original shapes and speeds completely intact.

It's as if they were solid particles bouncing off each other, which is why they are called solitons (the "-on" suffix is often used for particles, like protons or electrons). The only change after the collision is a slight shift in their position compared to where they would have been if they hadn't collided.

You can see this particle-like behaviour for yourself in our interactive KdV soliton simulation.