Convective Turing bifurcation

Following the approach pioneered by Eckhaus, Mielke, Schneider, and others for reaction–diffusion systems, we justify rigorously by Lyapunov–Schmidt reduction the formal amplitude (complex Ginzburg–Landau) equations describing Turing-type bifurcations of general reaction–diffusion–convection systems, showing that small spatially periodic traveling wave solutions of the PDE lie asymptotically close to spatially periodic traveling waves of the amplitude equations, with asymptotically nearby speeds. Notably, our analysis includes also higher-order, nonlocal, and even certain semilinear hyperbolic systems. This is the first step in a larger program, laying the groundwork for spectral stability analysis, and, ultimately, treatment of systems possessing conservation laws.