A direct theory for the perturbed unstable nonlinear Schrodinger equation

A direct perturbation theory for the unstable nonlinear Schrodinger 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 4x4 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. (C) 2000 American Institute of Physics. [S0022- 2488(00)00405-9].