Instability of Plane Poiseuille Flow Caused by a Nonlinear, Nonperiodic and Three-Dimensional Disturbance

By means of the Fourier transform, an amplitude expansion and a wavenumber expansion, the Navier-Stokes equation is reduced to a weakly nonlinear equation for the slowly varying complex amplitude of an envelope of a quasi-monochromatic and weakly three-dimensional disturbance. The reduced equation for the amplitude discloses severe condition of justification of the nonlinear Schrödinger type equation. The weak three-dimensionality of disturbance as well as the weak nonperiodicity has significant effects on the instability of plane Poiseuille flow.