Pattern Formation in Systems with Slowly Varying Geometry

A class of equations is considered which describe the evolution of patterns in problems with a geometry varying slightly, as characterized by a small parameter ν 1, and slowly, on a spatial scale ε x for ε 1. The model, inspired by the problem of sedimentation in meandering rivers, is generalized for an analytical study of nonlinear dynamics of periodic and quasi-periodic solutions which bifurcate from a basic solution. Two cases prove to be of interest: slightly, slowly varying geometry and slightly, very slowly varying geometry. Nonlinear modulation equations are studied foreach case, yielding analytical results complimentary to previous linear and numerical analyses. For slightly, slowly varying geometry the basic solution loses its stability to quasi-periodic solutions whose existence and stability are studied in the context of the nonlinear modulation equation. In the slightly, very slowly varying case there are stationary spatially periodic solutions of the modulation equation for all x for values R > Rₘₐₓ and no such solutions for R < Rcr. In an interval Rcr < R < Rₘₐₓ there are stable localized structures for which a uniform asymptotic approximation is constructed analytically.