  Generalized solitary waves with exponentially small non-decaying far field
oscillations have been studied in a range of singularly-perturbed differential
equations, including higher-order Korteweg-de Vries (KdV) equations. Many of
these studies used exponential asymptotics to compute the behaviour of the
oscillations, revealing that they appear in the solution as special curves
known as Stokes lines are crossed. Recent studies have identified similar
behaviour in solutions to difference equations. Motivated by these studies, the
seventh-order KdV and a hierarchy of higher-order KdV equations are
investigated, identifying conditions which produce generalized solitary wave
solutions. These results form a foundation for the study of infinite-order
differential equations, which are used as a model for studying lattice
equations. Finally, a lattice KdV equation is generated using finite-difference
discretization, in which a lattice generalized solitary wave solution is found.
