What is the difference between solutions of the diffusion equation with an imaginary diffusion coefficent and the wave equation's?

The diffusion equation of the form:

$$ \frac{\partial u(x,t)}{\partial t} = D\frac{\partial ^2u(x,t)}{\partial x^2} $$

If one chooses a real value for $D$, the solutions are usually decaying with time.

However, in some situations in physics, most notably the time-dependent Schrödinger equation, one sees an equation of similar form to the diffusion equation, but with a complex diffusion coefficient, i.e $D=i\,D'$.

This causes the equation's solutions to osculate instead of decay with time because

$$ \exp(-Dt)=\exp(-iD't) $$

Which is why the Schrödinger equation has wave solutions like the wave equation's.

It seems like one can transform the diffusion equation to an equation that can replace the wave equation since the solutions are the same.

This does not make much intuitive sense to me, so I think my understanding of the solutions of the wave and diffusion equation is not complete. What is the difference, if any, in the set of solutions of the diffusion equation with an imaginary diffusion coefficient and the wave equation's?

Answer

Both Schrödinger and Wave Equation have plane wave solutions, that's right. The difference is the dispersion relation, which is quadratic for the Schrödinger equation and linear for the wave equation. This is important, because the Schrödinger equation was designed to correctly reproduce the quadratic dispersion relation that was observed for electrons.

(You can show that by fourier-transforming both equations both in space and time and solving for $\omega(k)$.)