A Nehari manifold optimization method (NMOM) is introduced for finding 1-saddles, i.e., saddle points with the Morse index equal to one, of a generic nonlinear functional in Hilbert spaces. Actually, it is based on the variational characterization that 1-saddles of this functional are local minimizers of the same functional restricted on the associated Nehari manifold. The framework contains two important ingredients: one is the retraction mapping to make the iterative points always lie on the Nehari manifold; the other is the tangential search direction to decrease the functional with suitable step-size search rules. Particularly, the global convergence is rigorously established by virtue of some crucial analysis techniques (including a weak convergence method) that overcome difficulties in the infinite-dimensional setting. In practice, combining with an easy-to-implement Nehari retraction and the negative Riemannian gradient direction, the NMOM is successfully applied to compute the unstable ground-state solutions of a class of typical semilinear elliptic PDEs, such as the stationary nonlinear Schr\"odinger equation and the H\'enon equation. In particular, the symmetry-breaking phenomenon of the ground states of the H\'enon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of the symmetry-breaking reported.

翻译：本文提出了一种Nehari流形优化方法（NMOM），用于寻找希尔伯特空间中一般非线性泛函的1-鞍点（即莫尔斯指数为1的鞍点）。该方法基于如下变分特征：该泛函的1-鞍点即为该泛函限制在相应Nehari流形上的局部极小值点。该框架包含两个核心要素：一是通过回缩映射确保迭代点始终位于Nehari流形上；二是通过切向搜索方向结合合适的步长搜索规则实现泛函值的下降。特别地，通过运用若干关键分析技术（包括弱收敛方法）克服无限维空间中的困难，我们严格建立了算法的全局收敛性。在实际应用中，结合易于实现的Nehari回缩映射与负黎曼梯度方向，NMOM成功用于计算一类典型半线性椭圆型偏微分方程（如稳态非线性薛定谔方程和Hénon方程）的不稳定基态解。特别地，我们在一维和二维情况下数值探索了Hénon方程基态解的对称破缺现象，并报告了关于对称破缺临界值的有趣数值发现。