Bifurcation of Nonclassical Viscous Shock Profiles from the Constant State

We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.