Differential equations and non-linearity

The way most of us learn scientific disciplines is by the use of well established formulas (something like the ideal gas law relating the pressure, temperature and volume, PV=nRT). It is likely true that many scientific disciplines are vitally dependent on such "pure facts", however it is also true that any active discipline will consist of many findings that are considerably more controversial. After years of repetitive learning, many students find this unsettling. Fluid mechanics has a few well known formulas found in elementary physics texts (i.e. Bernoulli's equation), but for the most part, the "flowing" nature of the problems requires differential equations for a valid description. This is because differential equations relate the value of a function to its rate of change with time. While the laws of fluid mechanics in the form of the Navier-Stokes equations are partial differential equations (the quantities one wants to solve for are functions of more than one variable) here I discuss some general aspects of differential equations.

Perhaps the simplest differential equation is d/dt A(t) = k A(t), the solution of which defines the family of exponentials, A(t)= C exp(kt). Exponentials are generally taught quite early in the mathematics curriculum, but their peculiar properties only truly make sense when treated as the single example of a function for which the derivative is proportional to the function itself. This is the first way one can use differential equations: to define a new family of functions that one does not have a formula for ahead of time. The differential equation governing exponentials, like many other simple DEs (the simple harmonic oscillator, for one), is linear. This means if A_​1(t) and A_​2(t) are solutions, then     A_​1(t)+A_​2(t) is also a solution. Much like linear functions (lines) are the best understood functions of one variable, linear equations (be they differential or not) are the best understood equations. Indeed it is not that far off the mark to say that two hundred years of applied mathematics was largely concerned with deriving and solving linear equations. It may be surprising that a great many physical phenomena can be described by linear equations (most notably Quantum Mechanics is a fundamentally linear theory). We note that as computation has grown both in power and availability, more and more nonlinearity has entered into the discourse of modern applied mathematics.

At its most basic nonlinearity means we can no longer take two solutions and add them to produce a third. However, this is a purely mathematical road block, and nonlinearity would be a footnote were it not for the following two points:

1. Non-linearity enriches the possible dynamics of a system. On the one hand it can lead to disorder, as in the notion of "chaos" popularised in the late 1980s. A chaotic system can have two repetitions of the same experiment start very close to one another, but drift apart very quickly (indeed exponentially). However nonlinearity can also serve to preserve order. For linear waves an initial shape breaks up into a great many small ripples of varying lengths, each represented by a sine or cosine. For nonlinear waves, however, it is possible that an initial shape breaks up into a train of highly stable waves (called solitons) that propagate without changing form and interact as if they were particles (think pool balls) instead of waves.
2.  Non-linearity is a natural part of physical theories. In a flowing fluid the analogue of Newton's second law tells us how the velocity changes at a point. Some of the change must be due to the momentum carried by fluid that will arrive at the point an instant from now. In turn, this momentum, by definition depends on the velocity of the fluid. Of course the mathematics here is complicated (AMATH 231 is a good start, with AMATH 361 finishing the job). 

In physical systems the balance between order and disorder is much more robust than in simplified mathematical systems. Thus for example, a wind has a persistent direction even it has smaller variations (that cause a flag to flutter, for example). Physical systems tend to be forced with typical timescales (nights are cooler than days, spring is wetter than summer, the earth's orbit varies on a time scale of tens of thousands of years causing ice age cycles). Thus non-linearity in physics is often a balancing act between nearly random aspects and stable patterns.