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

Spectral stability analysis for solitary waves is developed in the context of the Hamiltonian system of coupled nonlinear Schrödinger equations. The linear eigenvalue problem for a non–self–adjoint operator is studied with two self–adjoint matrix Schrödinger operators. Sharp bounds on the number and type of unstable eigenvalues 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.