The amplitude equation for degenerate subcritical bifurcations in pattern formating systems

We consider a model for a pattern formating system on the infinite line which leads to a subcritical bifurcation. For the degenerate case we use multiple scaling analysis to derive as an amplitude equation for the envelope A of the bifurcating pattern which is modulated slowly in time and in space. We show exact estimates between the approximations obtained via the amplitude equation and true solutions of the original system. Moreover, we show that every small solution of the original system develops in such a way that it can be described after a certain time by the solutions of the amplitude equation. The difficulty is to show the estimates on an 𝓞(1/δ⁴)-time scale in contrast to 𝓞(1/δ²) for the classical Ginzburg-Landau equation, if δ is the order of the amplitude. This theory allows the description of modulated N-pulses in the original system.