Is the travelling wave ansatz the only solution for linear PDEs?

Question

If I have a linear PDE such as $$U_t=U_{xxx}+U_{xx}+U,$$ I know that particular solutions exist of the form $U=e^{\lambda t}e^{\alpha x}$. I heard that the total solution is the sum of such travelling waves. However can't other types of solutions exist? I often hear textbooks say they "seek" travelling wave solutions or that the solution is an ansatz, implying that other types exist. If not, what's the proof?

Furthermore if I have coupled equations such as

\begin{align} U_t&=U_{xx}+V_x+V\\ V_t&=V_{xx}+U_x, \end{align}

is the solution that $U,V$ are travelling waves also the only type of solution?

Answer 1

I would like to add it is only, as you mention, an ansatz. Others are possible and may lead to other solutions. For example, in the 3D scalar wave equation $$\frac{\partial^{2} f}{\partial t^{2}}\;=\; \nabla^{2} f$$ where $f\equiv f(t,x,y,z) $ the natural ansatz is a separable solution, but $$f(t,x,y,z) \;=\; \frac{1}{x^2+y^2+[A+i(z-t)][B-i(z+t)]}$$ is a (rather remarkable in my mind!) non-separable solution to the differential equation, where $A, B$ are constants.

Answer 2

No, the travelling wave solutions are not the only kind. Since the PDE is linear, any linear combination of travelling wave solutions is itself a solution, and such superpositions need not be travelling waves. For instance, both $e^{ix}e^{it}$ and $e^{-ix}e^{-it}$ are solutions of the linear wave equation $U_{xx}+U_{tt}=0$. Consequently, $e^{ix}e^{it}+e^{-ix}e^{-it}=2\cos(x+t)$ is also a solution but is not of the form $e^{\lambda t}e^{\alpha x}$. The same idea applies to a linear system of such PDEs.