Stability criterion for multicomponent solitary waves

We obtain the most general matrix criterion for stability and instability of multicomponent solitary waves by considering a system of N incoherently coupled nonlinear Schrödinger equations. Soliton stability is studied as a constrained variational problem which is reduced to finite-dimensional linear algebra. We prove that unstable (all real and positive) eigenvalues of the 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 solutions.