Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms

The problem of dimension breaking, for gradient elliptic partial differential equations in the plane, from a family of one-dimensional spatially periodic patterns (rolls) is considered. Conditions on the family of rolls are determined that lead to dimension breaking in the plane governed by a KdV equation relative to the periodic state. Since the KdV equation is time-independent, the -pulse solutions of KdV provide a sequence of multi-pulse planforms in the plane bifurcating from the rolls. The principal examples are the nonlinear Schrodinger equation, with evolution in the plane, and the steady Swift-Hohenberg equation with weak transverse variation.