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 the generic 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 iteration points always lie on the Nehari manifold; the other is the tangential search direction to decrease the generic 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) overcoming 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 H\'enon equation and the stationary nonlinear Schr\"odinger equation. In particular, the symmetry-breaking phenomenon of the ground states of H\'enon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of symmetry-breaking reported.

翻译：提出了一种Nehari流形优化方法（NMOM），用于寻找Hilbert空间中一般非线性泛函的1-鞍点，即Morse指标为1的鞍点。该方法基于如下变分刻画：一般泛函的1-鞍点是该泛函在关联Nehari流形上的局部极小点。该框架包含两个关键要素：其一是使迭代点始终位于Nehari流形上的回缩映射；其二是通过合适的步长搜索规则降低一般泛函的切向搜索方向。特别地，借助若干关键分析技巧（包括弱收敛方法），严格建立了全局收敛性，克服了无穷维框架下的困难。在实际应用中，结合易于实现的Nehari回缩与负黎曼梯度方向，NMOM成功用于计算一类典型半线性椭圆型偏微分方程（如Hénon方程和定态非线性Schrödinger方程）的非稳基态解。特别地，利用该方法对一维和二维情形下Hénon方程基态的对称破缺现象进行了数值探索，并报告了关于对称破缺临界值的有趣数值发现。