  We develop modulation theory for undular bores (dispersive shock waves) in
the framework of the Gardner, or extended Korteweg--de Vries, equation, which
is a generic mathematical model for weakly nonlinear and weakly dispersive wave
propagation, when effects of higher order nonlinearity become important. Using
a reduced version of the finite-gap integration method we derive the
Gardner-Whitham modulation system in a Riemann invariant form and show that it
can be mapped onto the well-known modulation system for the Korteweg--de Vries
equation. The transformation between the two counterpart modulation systems is,
however, not invertible. As a result, the study of the resolution of an initial
discontinuity for the Gardner equation reveals a rich phenomenology of
solutions which, along with the KdV type simple undular bores, include
nonlinear trigonometric bores, solibores, rarefaction waves and composite
solutions representing various combinations of the above structures. We
construct full parametric maps of such solutions for both signs of the cubic
nonlinear term in the Gardner equation. Our classification is supported by
numerical simulations.
