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Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, andTexier and Zumbrun, we propose as a mechanism for the onset of cellularinstability of viscous shock and detonation waves in a finite-cross-sectionduct the violation of the refined planar stability condition of Zumbrun--Serre,a viscous correction of the inviscid planar stability condition of Majda. Moreprecisely, we show for a model problem involving flow in a rectangular ductwith artificial periodic boundary conditions that transition tomultidimensional instability through violation of the refined stabilitycondition of planar viscous shock waves on the whole space generically impliesfor a duct of sufficiently large cross-section a cascade of Hopf bifurcationsinvolving more and more complicated cellular instabilities.  The refined condition is numerically calculable as described inBenzoni-Gavage--Serre-Zumbrun.

The refined inviscid stability condition and cellular instability of viscous shock waves