Nullspace-preserving high-index saddle dynamics method for degenerate multiple solution problems

TL;DR

NPHiSD addresses the challenge of locating high-index generalized saddle points in degenerate energy landscapes by performing nullspace–preserving ascent along a fixed complementary subspace on segments, thereby enabling upward searches that were previously hindered by Hessian nullspaces. The method combines unconstrained and sphere-constrained formulations, a segment-wise feasibility criterion based on $C_{\Theta}$ and $\rho_k$, and both explicit and semi-implicit time discretizations with proven stability and error bounds. The approach is demonstrated on three representative systems—the Lifshitz-Petrich model, the Gross-Pitaevskii energy, and Lennard-Jones clusters—where it produces a coherent solution landscape and identifies high-index parent states for downward searches. The results show NPHiSD to be universal, robust, and computationally efficient for degenerate problems, with potential applications to quantum systems, superconductivity, and skyrmions.

Abstract

We propose the nullspace-preserving high-index saddle dynamics (NPHiSD) method for degenerating multiple solution systems in constrained and unconstrained settings. The NPHiSD efficiently locates high-index saddle points and provides parent states for downward searches of lower-index saddles, thereby constructing the solution landscape systematically. The NPHiSD method searches along multiple efficient ascent directions by excluding the nullspace, which is the key for upward searches in degenerate problems. To reduce the cost of frequent nullspace updates, the search is divided into segments, within which the ascent directions remain orthogonal to the nullspace of the initial state of each segment. A sufficient and necessary condition for characterizing the segment that admits efficient ascent directions is proved. Extensive numerical experiments for typical problems such as Lifshitz-Petrich, Gross-Pitaevskii, and Lennard-Jones models are performed to show the universality and effectiveness of the NPHiSD method.

Figures (5)

• Figure 1: A simple illustration of $v_j$, $u^c_j$ and $\theta_j^c$ for $j=1$.
• Figure 1: Stable states of the Lifshitz-Petrich model, where the subgraphs show their spectral points in the reciprocal space (the arrows present the primitive reciprocal vectors).
• Figure 2: Solution landscape of crystal (with energy for each solution) where the GSPi (i = 1, 2) represents the index-i GSP. The critical nuclei are marked by white curves. The blue arrow represents the upward search by NPHiSD, and the block arrows represent the downward searches by HiSD (similar notations will be used subsequently).
• Figure 3: Vortex landscape of Bose-Einstein condensate (with energy for each solution). Here, $C$, $E$, $V$, and $G$ represent the centering, edge, and vertex vortices and the ground state, respectively, and the subscript indicates the number of vortices.
• Figure 4: Solution landscape of Lennard-Jones clusters with seven particles (with energy for each state).

Theorems & Definitions (11)

• Remark 3.1
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 4.1
• Remark 4.1
• Proof 3
• Remark 4.2
• Lemma 5.1