Diffusive-dispersive traveling waves and kinetic relations. Generic flux with multiple inflection points

We consider the broad class of scalar hyperbolic conservation laws with generic non-convex fluxes, which may thus have an arbitrary finite number of inflection points. We investigate the properties of classical and nonclassical shock waves realized as traveling-wave limits of a linear diffusive-dispersive regularization and provide a phase-plane classification beyond the case of a single inflection point. Within the speed range of the generalized Maxwell construction, our main result classifies the left outgoing profiles by their limiting equilibria. In an elementary bowtie configuration, we prove the existence of nonclassical shock profiles with arbitrarily many oscillations. For nested configurations, we order the saddle connections on each entry interval, taking account of inaccessible saddles. We show that the number of oscillations present in the non-classical shock depends on the diffusion-dispersion ratio, and tends to infinity along the elementary families of saddle connections as this ratio tends to zero. Our construction is based on a generalization of Maxwell's equal area rule, which provides us with an invariant region in the phase space for the study of diffusive-dispersive traveling waves.