Abstract

The nature of wave interaction in a continuum dynamical model may undergo a qualitative change in certain asymptotic regimes, most notably when linearity or complete integrability is introduced. This occurs in particular when the mKdV equation is used to model the unidirectional dispersive dynamics of two layer shallow water fluid flow near a critical interfacial height. Motivated by the symmetric properties of conjugate states which have been observed for the MCC equations in the Boussinesq limit, the work to be presented elucidates a more subtle qualitative shift, residing purely in the dispersionless reduction of a system, which determines whether a Hamiltonian undercompressive shock, representing a kink, will interact with a gradual background wave without producing a loss of regularity, which would take the form of a classical dispersive shock. This property is also related to an infinitude of conservation laws, drawing a further parallel to the integrable case.  This is a joint work with Roberto Camassa and Lingyun Ding.