How can one tell if a PDE describes wave behaviour?

Question

I have been looking at a lot of different non-linear PDEs which describe waves lately and have come to the realisation that I don't know what it is about these PDEs that make them behave like waves. For example, the Schrodinger equation, $i u_{t}+u_{xx}=0$ describes wave behaviour however the very similar looking diffusion equation $u_t-u_{xx}=0$ doesn't. What is it that makes the KdV equation, the CHM equation, and the Nonlinear Schrodinger equation exhibit wave behaviour?

Answer 1

The problem with answering this question is defining exactly what we mean by a wave and wave-behavior. I will here describe some properties waves have and show some examples of PDEs which describe different forms of wave behavior. I'll also touch upon some different ways we can define wave-behavior for linear and nonlinear PDEs.

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The simplest forms of what we call waves is solution of a PDE on the form $u(x,t) = f(x\pm ct)$. This represents a disturbance travelling with velocity $c$ to the right (-) or left (+). Such a solution also satisfy the wave-equation $u_{tt} = c^2u_{xx}$ and has the property that it does not dissipate energy and there is no dispersion. For example the linear Schrödinger equation $iu_t = u_{xx}$ has (periodic) wave-solutions $e^{i(x+t)} = \cos(x+t) + i\sin(x+t)$. However not all of such solutions are waves in the intuitive sense. For example $f(x) = e^x \implies u(x,t) = e^{x+t}$ can be seen as a wave under the definition above. Note that this is a solution to the diffusion equation which you don't see as having wave behavior. However if we also impose that the PDE allow solutions when $f$ is localized (see wave-packets) or periodic then when we get closer to the intuitive understanding most people have of a wave.

The definition above does not represent all waves as it does not allow for dissipation and dispersion (in the case where we go beyond a single plane wave). For example the PDE $u_t + u_x = u_{xx}$ has wave solutions $u(x,t) = \cos(k(x-t))e^{-k^2t}$. The last factor describes the decrease in amplitude of the wave as it travels and because of this it can not be written on the form $f(x-ct)$ as above. Dispersion means waves of different frequency have different velocity. A simple example is the PDE $u_t + u_x + u_{xxx} = 0$ which has solutions $\cos(k\left(x-\frac{\omega}{k}t\right))$. If there is no dispersion (as for the wave-equation) we have $c = \frac{\omega}{k} = $ constant while for the equation here we have $c(k) = \frac{\omega}{k} = 1 - k^2$ thus waves with larger wavelengths travel faster than waves of shorter wavelengths.

How to define wave-behavior for non-linear PDEs is more messy. Linear PDEs satisfy the superposition principle which says that any solution can be seens as a superposition of simpler solutions (like e.g. $\cos(kx-\omega(k) t)$ waves) so if we can show a PDE has some simple wave-solutions then we can build up any solution as a superposition of these waves. This is not true for non-linear PDEs. However nonlinear PDE's can have other types of waves known as solitons. A soliton is a self-reinforcing solitary wave, that has permanent shape, is localised within a region, does not satisfy the superposition principle, propagates at a constant velocity and does not disperse. Physically, such waves are caused by a cancellation of nonlinear and dispersive effects. All the nonlinear equations you mention as having wave-behavior has solitons so this is one possible answer to your question (they have wave-behavior because they have solitons). For example the Korteweg–de Vries (KdV) equation $u_t + u_{xxx} + 6 u u_x = 0$ has the soliton $u(x,t) = \frac{c}{2}\text{sech}\left(\frac{\sqrt{c}}{2}(x-ct)\right)$. The same is true for the nonlinear Schrödinger equation and the Charney-Hasegawa-Mima (CHM) equation.