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Based on a revised version of inverse scattering transform for the derivative nonlinear Schrödinger (DNLS) equation with vanishing boundary condition (VBC), the explicit N -soliton solution has been derived by some algebra techniques of some special matrices and determinants, especially the Binet-Cauchy formula. The one-and two-soliton solutions have been given as the illustration of the general formula of the N -soliton solution. Moreover, the asymptotic behaviors of the N -soliton solution have been discussed.

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A direct perturbation theory for the unstable nonlinear Schrödinger equation with perturbations is developed. The linearized operator is derived and the squared Jost functions are shown to be its eigenfunctions. Then the equation of linearized operator is transformed into an equivalent 4ϫ4 matrix form with first order derivative in t and the eigenfunctions into a four-component row. Adjoint functions and the inner product are defined. Orthogonality relations of these functions are derived and the expansion of the unity in terms of the four-component eigenfunctions is implied. The effect of damping is discussed as an example.