  We study periodic, two-dimensional, gravity-capillary traveling wave
solutions to a viscous shallow water system posed on an inclined plane. While
thinking of the Reynolds and Bond numbers as fixed and finite, we vary the
speed of the traveling frame and the degree of the incline and identify a set
of the latter two parameters that classifies from which combinations nontrivial
and small amplitude solution curves originate. Our principal technical tools
are a combination of the implicit function theorem and a local multiparameter
bifurcation theorem. To the best of the author's knowledge, this paper
constitutes the first construction and mathematical study of properly two
dimensional examples of viscous roll waves.
