Stability criterion for multicomponent solitary waves

We obtain the most general matrix criterion for stability and instability o f multicomponent solitary waves by considering a system of N incoherently c oupled nonlinear Schrodinger equations. Soliton stability is studied as a c onstrained variational problem which is reduced to finite-dimensional linea r algebra. We prove that unstable (all real and positive) eigenvalues of th e linear stability problem for multicomponent solitary waves are connected with negative eigenvalues of the Hessian matrix. The latter is constructed for the energetic surface of N-component spatially localized stationary sol utions.