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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:281

Article

Chen, Xianjin¹  Zhou, Jianxin² 

[1] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Peoples R China

[2] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

关键词: Cooperative elliptic system;    Multiple solutions;    Local min-orthogonal method;    Saddle point;    Spatial vector soliton;   

DOI  :  10.1016/j.cam.2014.11.046

来源: Elsevier

【 摘 要 】

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. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free