On steady surface waves over a periodic bottom: Relations between the pattern of imperfect bifurcation and the shape of the bottom

A bifurcation problem for steady surface waves over a periodic bottom is studied. It is assumed that the motion of the fluid is symmetric with respect to a vertical axis and periodic in the horizontal direction. In the case of flat bottom, we have infinitely many bifurcations from the trivial uniform flow. All of them are the pitchfork bifurcation and occur subcritically. If the bottom is not flat but close to flat, then the corresponding bifurcation equation is subject to a small perturbation. Since we know a universal unfolding of the pitchfork, the bifurcation diagram must be equivalent to one of several particular patterns. We will give relations between the patterns and functions representing the bottom, that is, we will specify which pattern is realised.