In 1834, a Scottish engineer named John Scott Russell witnessed something he did not expect: a single wave rolling down a narrow canal without changing its shape. Intrigued, he chased it on horseback and later described it as a “wave of translation.” At a time when waves were thought to quickly spread out and disappear, Russell’s moving bump of water seemed almost impossible. His observation raised a lasting question: how can a wave travel long distances without breaking apart?

Decades later, this mystery found a mathematical explanation. In the late nineteenth century, Diederik Korteweg and Gustav de Vries wrote down an equation for shallow water waves that allowed such stable traveling waves to exist. Their equation showed that two competing effects, one that tries to spread the wave out and one that tries to steepen it, can balance perfectly.

The story did not end there. Over the twentieth century, mathematicians and physicists refined these ideas, searching for better ways to describe water waves and other wave phenomena. Important insights were provided by Gerald Whitham, who emphasized how simple wave patterns can slowly change as they move.

The journey continues into the 1990s with a new chapter written by Roberto Camassa and Darryl Holm. Their Camassa–Holm equation predicts solitary waves with sharp crests, called “peakons,” revealing that even stranger wave shapes can arise from the same shallow water setting that inspired Russell’s canal experiment.

This talk follows the path from a man on horseback chasing a wave to modern mathematical equations, illustrating how a simple physical curiosity grew into a rich theory of solitary waves and nonlinear equations.