Travelling wave solution of a PDE

A function $u(x,t)$ is a travelling wave solution for a given PDE if it has the form

$$ u(x,t)=f(x-ct) $$

It is a fact that so many PDEs have solutions of this type, mainly the wave equation.

Example: The equation $u_t+cu_x=0$ admits any function of this type as a solution, as it is easy to check. This is the transport equation.

Example of non-existence: there are equations which don't have solutions of this kind. For example, the diffusion equation

$$ u_t=Du_{xx} $$

They are the similarity solutions associated to the symmetry

$$ c\partial_x+\partial_t $$

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Author of the notes: Antonio J. Pan-Collantes

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