Laura Pinkney (University of Leeds) – Pattern formation driven by three-wave interactions with two critical wavelengths

Three-wave interactions (3WIs) arise from a set of three wavevectors where the sum of first two gives the third one. We consider problems with two critical wavenumbers, where resonant triads form between two waves of a larger wavenumber and a third wave of a smaller wavenumber. 3WIs can be used to explain pattern-forming behaviour in the Faraday wave experiment close to onset. The experiment involves periodically forcing a container of fluid up and down and observing the patterns formed on the surface. When the forcing exceeds some threshold, the flat state becomes unstable, which can lead to a variety of patterns.

In this talk we will discuss the types of patterns formed by 3WIs with two wavenumbers and how these are different from patterns comprising a single wavelength. The dynamics exhibited by 3WIs can be represented as a system of complex ODEs, one equation for each amplitude of the wavevectors. We will consider how well this ODE system can predict the dynamics of a model PDE, an adaptation of the Lifschitz—Petrich equation.