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M. Chugunova and D. Pelinovsky

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Block-diagonalization of the symmetric first-order coupled-mode system

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SIAM Journal of Applied Dynamical Systems 5, 66-83 (2006)

**Abstract:**

We consider the Hamiltonian first-order coupled-mode system that
occur 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.

**Keywords**:

Hamiltonian first-order coupled-mode systems, gap solitons, spectral
stability, invariant subspaces, eigenvalues