AI 中文总结
研究不同内积下泛函\(F\)的鞍点动力学，建立鞍点与线性稳定稳态等价关系，给出离散内积动力学并验证收敛阶，应用于相场模型，揭示鞍点及连通性，凸显内积选择对保守系统解空间的丰富作用。
AI 中文摘要
为揭示内积选择的影响，我们给出了在不同内积下具有质量约束的泛函\(F\)的鞍点动力学统一公式。建立了指标为\(k\)的鞍点与相应动力学线性稳定稳态之间的等价关系。给出了离散\(H^{-1}\)和\(L^2\)内积的动力学，并通过数值验证了两种动力学的收敛阶。最后将该方法应用于具有驱动力的相场模型，在诺伊曼和周期边界条件下，结果揭示了先前未报道的鞍点及其连通性，突出了内积选择如何丰富保守系统中的解空间。
英文摘要
To reveal the effect of the inner product choice, we present a unified formulation of saddle dynamics for the functional F with a mass constraint under different inner products. We establish the equivalence between the index-k saddle points and the linearly stable steady states of the corresponding dynamics. Further, we present the dynamics with discrete H^{-1} and L^2 inner products and numerically verify the convergence orders of both dynamics. Finally, we apply the method to a phase field model with driving force under Neumann and periodic boundary conditions. The results uncover previously unreported saddle points and their connectivity, highlighting how the choice of inner product enriches the solution landscape in conservative systems.
Comments18 pages, 8 figures
