Modulational Instability, Turbulence, Collapse, And Inverse Cascade According To The Generalized Nonlinear Schr(?)dinger Equation

In this dissertation the modulation of wave fields, their condensation and collapse, as well as the resulting turbulence are investigated. The process is described by the two-dimensional complex nonlinear Schrodinger (CNLSE) or Ginsburg-Landau (CNLGLE) equation, which is now a paradigm equation describing nonlinear phenomena in many areas of physics, chemistry, and biology, as well as sociology and finance.In plasma physics, the NLSE was first used to describe plasma waves that are destabilized by the much lower frequency ion acoustic waves. The instability is to the enhancement of the latter by the ponderomotive force arising from the plasma waves. The enhanced ion acoustic waves in turn modulate the plasma waves, self-consistently leading to the so-called modulational instability. Since in plasmas, especially magnetized plasmas, there are many high and low frequency waves, such modulational instabilities can often appear. Zakharov in 1973 first predicted the phenomenon of wave collapse as a mathematical singularity (the local wave energy becomes infinite as the volume containing the wave fields goes to zero) of the plasma and ion acoustic wave system, as described by the NLSE. Since then there has been a huge number of studies on the latter and related equations. However, up to now no one has investigated the entire process from the modulational instability to the plasma wave turbulence after the collapse.Here we investigate the entire process using the generalized NLSE, which allows for complex coefficients, so that linear and nonlinear dissipation are included. Because of this generalization, the resulting complex NLSE, or CNLSE, becomes equivalent to the complex nonlinear Ginsburg-Landau equation, which also describes a wide class of physical problems in many areas.We solve the two-dimensional CNLSE numerically. Using different parameter values and initial conditions, we look for modulational instabilities that lead to turbulence. It is found that two pathways to turbulence are possible. One, without going through collapse, leads to a turbulent field which is relatively smooth. The other, which involves collapse, leads to a spiky turbulent field. The conditions and the processes are discussed.Clearly, there can exist other pathways that may lead to other types of turbulence or even self-organized coherent structures. All such processes are of interest to the problems described by the CNLSE, and thus worth further investigation.