Snakes, ladders, and isolas of localised patterns

Stable localised roll structures have been observed in many physical problems and model equations, notably in the 1D Swift–Hohenberg equation. Reflection-symmetric localised rolls are often found to lie on two “snaking” solution branches, so that the spatial width of the localised rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many “ladder” branches of asymmetric localised rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localised in one spatial direction.