  We introduce and study a family of lattice equations which may be viewed
either as a strongly nonlinear discrete extension of the Gardner equation, or a
non-convex variant of the Lotka-Volterra chain. Their deceptively simple form
supports a very rich family of complex solitary patterns. Some of these
patterns are also found in the quasi-continuum rendition, but the more
intriguing ones, like interlaced pairs of solitary waves, or waves which may
reverse their direction either spontaneously or due a collision, are an
intrinsic feature of the discrete realm.
