Page 3 model: Solitons



Solitons are special analytic solutions to the nonlinear wave equations that turn up everywhere: from fluid dynamics to quantum mechanics to molecular biology. One such equation is the Korteweg–de Vries (KdV) equation for a 1D wave $u(x,t)$ evolving in time $t$: \[ \frac{\partial u}{\partial t}+6u\frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0. \] What makes solitons special is that they behave in many ways like solutions to the linear wave equation: propagating without losing their shape, and even interacting nicely:

In 1990, Daisuke Takahashi and Junkichi Satsuma proposed a discrete model for solitons called the ball and box model. It consists of an infinite row of boxes, some of which contain a ball. A sequence of consecutive balls represents a wave. The pattern of waves after each time step is found by starting from the left and moving each ball to the next available empty box to its right.

A row of $n$ consecutive balls behaves like a single wave, and moves to the right at a constant speed $n$. Several such waves, so long as they start sufficiently separated, will interact (in a possibly messy way), and then disperse, maintaining their original shapes overall.

This model is fun to play with, but—surprisingly given its simplicity—it manages to capture a large amount of the interesting and unusual behaviour of the fluid waves modelled by the KdV equation.

David teaches and researches maths at King’s College London, with a focus on geometry. In his spare time he likes to make music and read poetry.

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