7 research outputs found
Constrained high-index saddle dynamics for the solution landscape with equality constraints
We propose a constrained high-index saddle dynamics (CHiSD) to search
index- saddle points of an energy functional subject to equality
constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax
framework, and its linear stability at the index- saddle point is proved. To
ensure the manifold properties, the CHiSD is numerically implemented using
retractions and vector transport. Then we present a numerical approach by
combining CHiSD with downward and upward search algorithms to construct the
solution landscape with equality constraints. We apply the Thomson problem and
the Bose--Einstein condensation as the numerical examples to demonstrate the
efficiency of the proposed method
Solution landscapes of the simplified Ericksen--Leslie model and its comparison with the reduced Landau--de Gennes model
We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes of the corresponding rLdG models. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution with interior defects, high-index "fake defects" solutions, novel tessellating solutions, and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics