The discovery of solitons, and more generally nonlinear waves, created several new fields of study in mathematics, fluid dynamics, optics, and engineering. It is, undoubtedly, a fine example of applied mathematics. The discovery of nonlinear waves came through a combination of physical experiments, numerical analysis, and mathematical modelling. These results created a paradigm shift: once the initial structure was understood, the field rapidly expanded through mathematical and numerical analyses. It also influenced the design of new experiments and the modelling of other physical and biological systems.

A possible way to think about various theories of applied mathematics is to look at both the objects under study and their behaviour. Objects can be discrete (such as particles or planets) or continuous (such as pressure or temperature). Similarly, their evolution of behaviour can be discrete (as the sequences of heads or tails when flipping a coin ) or continuous (as in the position of planets, or the motion of a piano string). In 1993, Christopher Zeeman, an influential and at times controversial British applied mathematician, proposed that mathematical theories could be classified by placing them in different boxes according to both their behaviour and the thing that they model, as shown in Figure 20. We already recognize a few boxes. The time box contains discrete objects that behave continuously in time and was investigated

The continuous box is the classical field theories where continuous objects (fields) behave continuously in time. The interesting box for our discussion is Pandora’s box. Here, we discover new objects whose behaviour is unexpected and often sparks controversy. The prime example in this box is quantum theory with its duality between (continuous) waves and (discrete) particles. But the box also contains more delightful surprises: everyday objects that should behave continuously according to the 19th‐century theory, but actually behave like discrete particles. Solitons are one example of the treasures to be found here. They are all beautiful and mathematically exciting and invariably useful.

A box classification of applied mathematics in terms of objects and their behaviour. Examples for each box are given (top) as well as the theories they generate (bottom).