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We study the solution landscape and bifurcation diagrams of nematic liquidcrystals confined on a rectangle, using a reduced two-dimensional Landau--deGennes framework in terms of two geometry-dependent variables: half short edgelength $\lambda$ and aspect ratio $b$. First, we analytically prove that, forany $b$ with a small enough $\lambda$ or for a large enough $b$ with a fixeddomain size, there is a unique stable solution that has two line defects on theopposite short edges. Second, we numerically construct solution landscapes byvarying $\lambda$ and $b$, and report a novel X state, which emerges fromsaddle-node bifurcation and serves as the parent state in such a solutionlandscape. Various new classes are then found among these solution landscapes,including the X class, the S class, and the L class. By tracking the Morseindices of individual solutions, we present bifurcation diagrams for nematicequilibria, thus illustrating the emergence mechanism of critical points andseveral effects of geometrical anisotropy on confined defect patterns.

Nematic liquid crystals in a rectangular confinement: solution landscape and bifurcation