Propagation of nonclassical shock waves in Hall-magnetohydrodynamics model of non-ideal fluids

We investigate the behavior of nonlinear waves in the Hall-magnetohydrodynamics model of non-ideal fluids with the van der Waals equation of state. It is shown that the Hall effect leads to the formation of a nonclassical shock wave that does not follow Oleinik's entropy condition and propagates as an undercompressive shock. The evolution of a rectangular pulse is discussed by solving a Riemann problem with initial data containing a pair of discontinuities. In particular, we provide a thorough discussion of interaction of various waves emerging from the discontinuities. It is observed that during this interaction, a nonclassical shock may transform into a classical sonic shock depending on the critical value of the Hall parameter. Furthermore, we analyze the effects of van der Waals parameters and the Hall parameter on various aspects of wave propagation, such as shock speed, shock strength, width of wave fan, and interaction time.