Existence of Orthogonal Domain walls in Bénard-Rayleigh Convection

Abstract

In Bénard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x - coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in Buffoni et al. (J Diff Equ, 2023, https://doi.org/10.1016/j.jde.2023.01.026), and analytically in Iooss (Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection. Preprint, 2023). We then prove for a given amplitude \(\varepsilon ^2\), and an imposed symmetry in coordinate y, the existence of a one-parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked with an adapted shift of rolls parallel to the wall.