Note: this post was originally written by David Ketcheson.
Recently, a stackexchange answer claimed that the Schrodinger equation is effectively a reaction-diffusion equation. I’ll set aside semantic arguments about the meaning of “effectively”, and give a more obvious example to explain why I think this statement is misleading.
Consider the wave equation
\[u_{tt} = u_{xx}\]
Introducing a new variable \(v=u_t\) we can rewrite the wave equation as
\[ \begin{align*} v_t & = u_{xx} \\ u_t & = v. \end{align*} \]
Observe that the first of these equation is the diffusion equation, while the second is a reaction equation. Thus we have reaction-diffusion!
Right?
Wrong. We’ve disguised the true nature of this equation by applying our intuition (which is based on scalar PDEs) to a system of PDEs. In the same way, the “reaction-diffusion” label for Schrodinger is obtained by applying intuition based on PDEs with real coefficients to a PDE with complex coefficients.
Of course, in both cases you can use numerical methods that are appropriate for reaction-diffusion problems in order to solve a wave equation.
Here is a quick ipython notebook implementation of the obvious method for the system above.