A direct theory for the perturbed unstable nonlinear Schrodinger equation

A direct perturbation theory for the unstable nonlinear Schrodinger equatio n with perturbations is developed. The linearized operator is derived and t he squared Jost functions are shown to be its eigenfunctions. Then the equa tion of linearized operator is transformed into an equivalent 4x4 matrix fo rm with first order derivative in t and the eigenfunctions into a four-comp onent row. Adjoint functions and the inner product are defined. Orthogonali ty relations of these functions are derived and the expansion of the unity in terms of the four-component eigenfunctions is implied. The effect of dam ping is discussed as an example. (C) 2000 American Institute of Physics. [S 0022- 2488(00)00405-9].