Transverse steady bifurcation of viscous shock solutions of a system of parabolic conservation laws in a strip

We derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic–parabolic model in a strip. This is related to “cellular instabilities” occurring in detonation and MHD. Our results extend to multiple dimensions the results of [1] on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results of [19] on transverse Hopf bifurcation, showing that the bifurcating time-periodic solution is in fact a spatially periodic traveling wave. Our technique consists of a Lyapunov–Schmidt type of reduction, which prepares the equations for the application of other bifurcation techniques. For the reduction in transverse modes, a general Fredholm alternative-type result is derived, allowing us to overcome the unboundedness of the domain and the lack of compact embeddings; this result applies to general closed operators.