Nonlinearity is a basic fact of life when modeling the majority of natural systems. In the theoretical literature it is often shrouded in mysterious notation and has words like chaos associated with it. Yet the distinction between linear and nonlinear systems can be understood fairly simply.
In the picture above on the left a small water wave (the light streak about a third from the left of the photo) has propagated from right to left past a growing plant. This plant has left the wave largely unchanged except for a small ring of waves generated near the plant and visible as to the right of it. To an excellent approximation the system can be thought of as a sum of the original wave that has moved past the plant and an even smaller ring of waves propagating circularly out from the plant.
In the picture on the right the story is very different. Here one wave is moving from the top right toward the shore (to the left of the picture edge) a different wave is propagating diagonally away from shore (it was generated by a large breaking wave sometime earlier). The two waves interact near the center of the photo. From the foam generated when the waves break, it is quite evident that the system does not simply act as a sum of the two waves.
Mathematically, nonlinearity means that if we find two separate solutions to a set of equations we no longer expect the sum of the solutions to be a solution. This makes nonlinear equations much more difficult to solve, because the superposition principle no longer applies. It also makes them much more interesting to study (for example see the interesting article on cnoidal waves).
Nonlinearity
