Exponential asymptotics of localised patterns and snaking bifurcation diagrams

Localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple-scales analysis beyond all orders. The model studied is the Swift-Hohenberg equation of nonlinear optics, which is equivalent to the classical Swift-Hohenberg equation with a quadratic and a cubic nonlinearity. Applying the asymptotic technique away from the Maxwell point first, it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow -type amplitude. In the vicinity of the Maxwell point, the beyond-all-orders calculation yields the “pinning range” of parameters where stable stationary fronts connect the homogeneous and periodic states. The full bifurcation diagram for localised patterns is then computed analytically, including snake and ladder bifurcation curves. This last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up- and a down-front. To this end, a third, super-slow spatial scale needs to be introduced, in which fronts appear as boundary layers. In addition, the location of the Maxwell point and the oscillation wave number of localised patterns are required to fourth-order accuracy in the oscillation amplitude.