  Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics.
We provide novel exact analytical solutions to this non-linear mean-field
approach, for the diffuse layer of ions in the vicinity of a planar or a
cylindrical macroion. While previously known solution are for homogeneously
charged objects, the cases worked out exhibit a modulated surface charge --or
equivalently surface potential-- on the macroion (wall) surface. In addition to
asymptotic features at large distances from the wall, attention is paid to the
fate of the contact theorem, relating the contact density of ions to the local
wall charge density. For salt-free systems (counterions only), we make use of
results pertaining to the two-dimensional Liouville equation, supplemented by
an inverse approach. When salt is present, we invoke the exact two-soliton
solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge
patterns, that are either localized or periodic in space. Without salt, the
electrostatic signature of a charge pattern on the macroion fades exponentially
with distance for a planar macroion, while it decays as an inverse power-law
for a cylindrical macroion. With salt, our study is limited to the planar
geometry, and reveals that pattern screening is exponential.

Soft Condensed Matter;Statistical Mechanics