Inertia law for spectral stability of solitary waves in coupled nonlinear Schrodinger equations

Spectral stability analysis for solitary waves is developed in the context of the Hamiltonian system of coupled nonlinear Schrodinger equations. The linear eigen-value problem for a non-self-adjoint operator is studied with two self-adjoint matrix Schrodinger operators. Sharp bounds on the number and type of unstable eigen-values in the spectral problem are found from the inertia law for quadratic forms, associated with the two self-adjoint operators. Symmetry-breaking stability analysis is also developed with the same method.