# Pattern formation and nonlinear waves close to a 1:1 resonant Turing and Turing--Hopf instability

## TL;DR

This paper analyzes the dynamics near simultaneous Turing and Turing--Hopf instabilities with 1:1 resonance, deriving amplitude equations, constructing solutions, and modeling pattern transitions in a coupled Swift--Hohenberg system.

## Contribution

It introduces a novel analysis of coupled pattern-forming systems near resonant instabilities, deriving amplitude equations and constructing explicit solutions.

## Key findings

- Existence of space-time periodic solutions.
- Construction of fast-traveling front solutions.
- Modeling of spatial pattern transitions.

## Abstract

In this paper, we analyse the dynamics of a pattern-forming system close to simultaneous Turing and Turing--Hopf instabilities, which have a 1:1 spatial resonance, that is, they have the same critical wave number. For this, we consider a system of coupled Swift--Hohenberg equations with dispersive terms and general, smooth nonlinearities. Close to the onset of instability, we derive a system of two coupled complex Ginzburg--Landau equations with a singular advection term as amplitude equations and justify the approximation by providing error estimates. We then construct space-time periodic solutions to the amplitude equations, as well as fast-travelling front solutions, which connect different space-time periodic states. This yields the existence of solutions to the pattern-forming system on a finite, but long time interval, which model the spatial transition between different patterns. The construction is based on geometric singular perturbation theory exploiting the fast travelling speed of the fronts. Finally, we construct global, spatially periodic solutions to the pattern-forming system by using centre manifold reduction, normal form theory and a variant of singular perturbation theory to handle fast oscillatory higher-order terms.