On the Nonlinear Schrödinger Equation for Langmuir Waves

A direct derivation of the nonlinear schrödinger equation for Langmuir waves is presented, based upon the nonlinear wave packet ansatz of Karpman and Krushkal. Both fluid and Vlasov equation formulations are used. The results obtained are essentially equivalent to those found earlier by Taniuti, et al. using reductive perturbation theory, including the importance of wave particle resonances at the group velocity for the long time behavior of the amplitude of modulated waves. Separating the wave packet considerations from the calculation of the nonlinear frequency shift makes it possible to attack the latter with whatever method facilitates the analysis of that part of the problem. In addition, certain ambiguities concerning singularities in velocity integrations are resolved, and the connection with a well-posed initial value problem is made somewhat clearer. This method can be used equally well for other waves, and may be of help particularly in situations where it is not clear, a priori, what scaling to adopt in applying reductive perturbation theory.