  The interaction of localised solitary waves with large-scale, time-varying
dispersive mean flows subject to nonconvex flux is studied in the framework of
the modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear
internal gravity wave propagation in stratified fluids. The principal feature
of the studied interaction is that both the solitary wave and the large-scale
mean flow -- a rarefaction wave or a dispersive shock wave (undular bore) --
are described by the same dispersive hydrodynamic equation. A recent
theoretical and experimental study of this new type of dynamic soliton-mean
flow interaction has revealed two main scenarios when the solitary wave either
tunnels through the varying mean flow that connects two constant asymptotic
states, or remains trapped inside it. While the previous work considered convex
systems, in this paper it is demonstrated that the presence of a nonconvex
hydrodynamic flux introduces significant modifications to the scenarios for
transmission and trapping. A reduced set of Whitham modulation equations,
termed the solitonic modulation system, is used to formulate a general,
approximate mathematical framework for solitary wave-mean flow interaction with
nonconvex flux. Solitary wave trapping is conveniently stated in terms of
crossing characteristics for the solitonic system. Numerical simulations of the
mKdV equation agree with the predictions of modulation theory. The developed
theory draws upon general properties of dispersive hydrodynamic partial
differential equations, not on the complete integrability of the mKdV equation.
As such, the mathematical framework developed here enables application to other
fluid dynamic contexts subject to nonconvex flux.
