Estimate of Morse index of cooperative elliptic systems and its application to spatial vector solitons

Local instability index of unstable solutions to single partial differential equations (PDEs) by a local minimax method (LMMM) was established in Zhou (2005). It is known that the local min-orthogonal method (LMOM) which was first proposed in Zhou (2004) and then further developed in Chen et al. (2008) can find more general unstable solutions to both single PDEs and cooperative elliptic systems. This paper is to carry out instability analysis of unstable solutions by LMOM, to which an infinite-dimensional functional space can be decomposed as a direct sum of a finite-dimensional subspace and its orthogonal complement. A Morse index approach is developed to show that with LMOM, instability behavior of a solution in such infinite-dimensional complement subspace can be totally determined. Usual instability analysis in an entire space is then reduced to analysis in its finite-dimensional subspace, for which a corresponding matrix decomposition is proposed to analyze a solution's instability behavior. Estimates of Morse index are also established. Finally, numerical examples of both 2- and 3-component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons, whose local instabilities are numerically confirmed by the new estimates. Certain important properties of the examples are also verified or presented.