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Abstract Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number$$F<2$$ $F<2$, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation. 

In an interesting recent analysis, Haragus–Johnson–Perkins–de Rijk have shown modulationalstability under localized perturbations of steady periodic solutions of the Lugiato–Lefeverequation (LLE), in the process pointing out a difficulty in obtaining standard “nonlinear dampingestimates” on modulated perturbation variables to control regularity of solutions. Here, we point outthat in place of standard “inverse-modulated” damping estimates, one can alternatively carry outa damping estimate on the “forward-modulated” perturbation, noting that norms of forward- andinverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classicalargument structure of Johnson–Noble–Rodrigues–Zumbrun for parabolic systems. This observationseems of general use in situations of delicate regularity. Applied in the context of (LLE), it gives thestronger result of stability and asymptotic behavior with respect to nonlocalized perturbations. 

We study for the Richard-Gavrilyuk model of inclined shallow water flow, an extension of the classical Saint Venant equations incorporating vorticity, the new feature of convective-wave solutions analogous to contact discontinuitis in inviscid conservation laws. These are traveling waves for which fluid velocity is constant and equal to the speed of propagation of the wave, but fluid height and/or enstrophy (thus vorticity) varies. Together with hydraulic shocks, they play an important role in the structure of Riemann solutions.