Block-Diagonalization of the Symmetric First-Order Coupled-Mode System

We consider the Hamiltonian first-order coupled-mode system that occurs in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupled-mode system, which is equivalent to a four-by-four Dirac system with sign-indefinite metric. In the special class of symmetric nonlinear potentials, we construct a block-diagonal representation of the linearized equations, when the spectral problem reduces to two coupled two-by-two Dirac systems. The block-diagonalization is used in fast numerical computations of eigenvalues with the Chebyshev interpolation algorithm.