  New types of stationary solutions of a one-dimensional driven sixth-order
Cahn-Hilliard type equation that arises as a model for epitaxially growing
nano-structures such as quantum dots, are derived by an extension of the method
of matched asymptotic expansions that retains exponentially small terms. This
method yields analytical expressions for far-field behavior as well as the
widths of the humps of these spatially non-monotone solutions in the limit of
small driving force strength which is the deposition rate in case of epitaxial
growth. These solutions extend the family of the monotone kink and antikink
solutions. The hump spacing is related to solutions of the Lambert $W$
function. Using phase space analysis for the corresponding fifth-order
dynamical system, we use a numerical technique that enables the efficient and
accurate tracking of the solution branches, where the asymptotic solutions are
used as initial input. Additionally, our approach is first demonstrated for the
related but simpler driven fourth-order Cahn-Hilliard equation, also known as
the convective Cahn-Hilliard equation.
