A rigorous justification of the Matthews-Cox approximation for the Nikolaevskiy equation

The Nikolaevskiy equation is an example of a pattern forming system with marginally stable long modes. It has the unusual property that the typical Ginzburg-Landau scaling ansatz for the description of propagating patterns does not yield asymptotically consistent amplitude equations. Instead, another scaling proposed by Matthews and Cox can be used to formally derive a consistent system of modulation equations. We give a rigorous proof that this system makes correct predictions about the dynamics of the Nikolaevskiy equation.