We study computational theory and methods for finding multiple unstable solutions
(corresponding to saddle points) to three types of nonlinear variational elliptic
systems: cooperative, noncooperative, and Hamiltonian. We first propose a new Lorthogonal
selection in a product Hilbert space so that a solution manifold can be
defined. Then, we establish, respectively, a local characterization for saddle points of
finite Morse index and of infinite Morse index. Based on these characterizations, two
methods, called the local min-orthogonal method and the local min-max-orthogonal
method, are developed and applied to solve those three types of elliptic systems for
multiple solutions. Under suitable assumptions, a subsequence convergence result
is established for each method. Numerical experiments for different types of model
problems are carried out, showing that both methods are very reliable and efficient in
computing coexisting saddle points or saddle points of infinite Morse index. We also
analyze the instability of saddle points in both single and product Hilbert spaces. In
particular, we establish several estimates of the Morse index of both coexisting and
non-coexisting saddle points via the local min-orthogonal method developed and propose
a local instability index to measure the local instability of both degenerate and
nondegenerate saddle points. Finally, we suggest two extensions of an L-orthogonal
selection for future research so that multiple solutions to more general elliptic systems
such as nonvariational elliptic systems may also be found in a stable way.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2990 |
| Date | 15 May 2009 |
| Creators | Chen, Xianjin |
| Contributors | Zhou, Jianxin |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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