Global-in-time existence of modulating front solutions near a 1:1 resonant Turing–Turing–Hopf instability

Establish the global-in-time existence of modulating front (pattern-interface) solutions for the coupled Swift–Hohenberg system of equations (1) that is analyzed near a 1:1 resonant Turing and Turing–Hopf instability. Specifically, construct solutions (u,v)(t,x) that connect distinct space–time periodic states as x→±∞ and persist for all t∈R, rather than only on long but finite time intervals obtained via amplitude-equation approximations, while rigorously handling the fast oscillatory higher-order terms arising from the different phase velocities and the inadequacy of the standard ansatz u(t,x)=U(x−ct,x−c_pt).

Background

The paper derives and justifies coupled complex Ginzburg–Landau amplitude equations for a system of two coupled Swift–Hohenberg-type equations close to a simultaneous 1:1 resonant Turing and Turing–Hopf instability. Using these amplitude equations, the authors construct fast-moving fronts that connect different space–time periodic states and translate them back to solutions of the full PDE, but only on long, finite time intervals determined by the multiscale expansion.

Achieving global-in-time fronts in the full PDE is substantially harder because the interaction of stationary (Turing) and traveling (Turing–Hopf) modes introduces highly oscillatory higher-order terms; moreover, the standard spatial-dynamics “modulating front” ansatz u(t,x)=U(x−ct,x−c_pt) is insufficient to capture these effects and fronts connecting patterns with different phase speeds.