Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scaling

We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed $c\sim {\epsilon }^{1/3}$, where $\epsilon$ is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of $c$, that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings–McLeod solution of the Painlevé-II equation to Painlevé-II equations with a drift term.