Linear stability of multiple dark solitary wave solutions of a nonlocal nonlinear Schrodinger equation for envelope waves

The linear stability analysis is performed for a new class of multiple dark solitary wave solutions of a nonlocal nonlinear Schrodinger (NLS) equation describing the long-term evolution of the envelope wave. The solutions to the linear eigenvalue problem associated with the inverse scattering transf orm for the nonlocal NLS equation are found explicitly for both continuous and discrete spectra and the completeness relation for the squared eigenfun ctions is proved. This allows us to solve the initial value problem of the nonlocal NLS equation Linearized about the solitary wave solutions and then establish their linear stability. (C) 2001 Elsevier Science B.V. All right s reserved.