A six-dimensional reversible normal form system occurs in Bénard–Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al. in J Diff Equ https://doi.org/10.1016/j.jde.2023.01.026, 2023). This leads to study analytically the system \(\begin{aligned} \frac{d^{4}A}{dx^{4}}= & A(1-A^{2}-gB^{2}) \\ \frac{d^{2}B}{dx^{2}}= & \varepsilon ^{2}B(-1+gA^{2}+B^{2}), \end{aligned}\) for \(x\in {\mathbb {R}},\) and looking for a heteroclinic connection between the two equilibria \( M_{-}:(A,B)=(1,0)\) and \(M_{+}:(A,B)=(0,1)\) , each corresponding to a system of convective rolls. In Buffoni et al. (J Diff Equ https://doi.org/10.1016/j.jde.2023.01.026, 2023) such a heteroclinic is shown to exist, on which \(0\le B\le 1,\) with no uniqueness result and no possibility to use it for a persistence result under a reversible perturbation. The lack of normal hyperbolicity in \((A,B)=(0,1/\sqrt{g})\) of equilibria obtained at the limit \(\varepsilon =0,\) is the main problem. The 3-dimensional unstable manifold of \(M_{-}\) is built for \(0\le B\le \frac{1-c\varepsilon ^{4/5}}{ \sqrt{g}},\) while, in solving a certain 4th-order differential equation independent of \(\varepsilon \) (occuring in Manneville and Pomeau (Phil Mag A 48(4): 607–621, 1983), Buffoni (On minimizers of an integral functional arising in the Bénard– Rayleigh convection problem, 2023)), we overcome the lack of hyperbolicity in building the stable manifold of \(M_{+}\) for \(\frac{1-c\varepsilon ^{4/5}}{\sqrt{g}}\le B\le 1.\) We use Buffoni et al. (J Diff Equ https://doi.org/10.1016/j.jde.2023.01.026, 2023) for proving that the two manifold intersect. Then the two 3-dimensional manifolds intersect transversally, leading to the existence, uniqueness and analyticity in \((\varepsilon ,g)\) of the heteroclinic, for which we give estimates of A(x), B(x) and their derivatives. We finally study the properties of the linearized operator along the heteroclinic, allowing to prove (in Iooss (J Math. Fluid Mech https://doi.org/10.1007/s00021-024-00891-2 (2024)) the persistence of the heteroclinic under perturbation, corresponding to the existence of orthogonal domain walls in the Bénard–Rayleigh convection problem.
