In this talk, we consider coupled nonlinear Schr\"odinger (CNLS) equations with a general nonlinearity. We assume that CNLS equations possess a solitary wave of which one component is identically zero and that the pitchfork bifurcation of this solitary wave occurs. Utilizing the Evans function approach, we locate eigenvalues of the linearized problem around the bifurcated solitary waves. Especially, embedded eigenvalues of negative (resp. positive) Krein sigunature become unstable (resp. disappear) after the bifurcation. These kinds of results were previously reported (e.g., [1,2]) but they were obtained by formal calculations and/or assuming some generic conditions. Our result gives a sufficient codition which seems easier to check. This talk is based on a joint work with K. Yagasaki.
 S. Cuccagna, D. Pelinovsky, and V. Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58 (2005), 1--29.
 D. Pelinovsky and J. Yang, Instabilities of multihump vector solitons in coupled nonlinear Schr\"odinger equations, Stud. Appl. Math., 115 (2005) 109--137.