Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913977401868288 |
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| author | Chen, Zhaoxing Liu, Wei Xie, Ziqing Yi, Wenfan |
| author_facet | Chen, Zhaoxing Liu, Wei Xie, Ziqing Yi, Wenfan |
| contents | 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 Schrödinger equation and the Hénon equation. In particular, the symmetry-breaking phenomenon of the ground states of the Hénon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of the symmetry-breaking reported. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_09892 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs Chen, Zhaoxing Liu, Wei Xie, Ziqing Yi, Wenfan Numerical Analysis 35B38, 58E30, 65K10, 65N12 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 Schrödinger equation and the Hénon equation. In particular, the symmetry-breaking phenomenon of the ground states of the Hénon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of the symmetry-breaking reported. |
| title | Nehari manifold optimization and its application for finding unstable solutions of semilinear elliptic PDEs |
| topic | Numerical Analysis 35B38, 58E30, 65K10, 65N12 |
| url | https://arxiv.org/abs/2404.09892 |