Bifurcation of nonclassical viscous shock profiles from the constant state

We determine the bifurcation from the constant solution of nonclassical tra nsitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation law s involve a single characteristic field, nonclassical waves involve two fie lds in an essential way. This feature is reflected in the viscous profile d ifferential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurca tion occurring in the classical case. We carry out a complete bifurcation a nalysis for systems of two quadratic conservation laws with constant, stric tly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a c odimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. On e consequence of our analysis is that weak transitional waves arise in pair s, with profiles forming a 2-cycle configuration previously shown to lead t o nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak no nclassical waves is necessarily associated with change of stability in cons tant solutions of the parabolic system of conservation laws, rather than wi th change of type in the associated hyperbolic system.