  We derive an asymptotic formula for the amplitude distribution in a fully
nonlinear shallow-water solitary wave train which is formed as the long-time
outcome of the initial-value problem for the Su-Gardner (or one-dimensional
Green-Naghdi) system. Our analysis is based on the properties of the
characteristics of the associated Whitham modulation system which describes an
intermediate "undular bore" stage of the evolution. The resulting formula
represents a "non-integrable" analogue of the well-known semi-classical
distribution for the Korteweg-de Vries equation, which is usually obtained
through the inverse scattering transform. Our analytical results are shown to
agree with the results of direct numerical simulations of the Su-Gardner
system. Our analysis can be generalised to other weakly dispersive, fully
nonlinear systems which are not necessarily completely integrable.
