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 (Formula presented.), where (Formula presented.) 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 (Formula presented.), 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.