This award will support research on critical nonlinear Schroedinger equations (NLE), i.e. equations with a cubic nonlinearity. Of interest are collapse events, that is, spatial contractions of solutions to single points in finite time. Individual collapse events are now well understood, and this work will study collapse turbulence, that is, solutions that exhibit random distributions of collapse events. For this purpose, the equation has to be regularized, since solutions cannot be continued beyond a complete collapse. One of the issues to be studied then is the dependence of solutions on the choice of regularization. It is conjectured that this regularization will only have a moderate effect on collapse turbulence, and this conjecture will be studied in this project. There is a general framework for the statistical study of turbulence in the context of the equations of fluid dynamics that goes back to Kolmogorov, and this work will place collapse turbulence of solutions of the NLS in this general framework. The topic is very suitable for graduate training, and students will be supported and exposed to work done by research groups at other universities and at national laboratories.
The nonlinear Schrodinger equation (NLS), which describes the nonlinear interaction of waves over time, is a universal model in nonlinear science. It occurs in the description of laser fusion, in fiber optics, and in models for rogue waves in oceanography. Stable moving waves (such as rogue waves or light pulses) are called solitons, and the spontaneous emergence of individual solitons in solutions of the NLS is now well understood. This work will study situations where such solitons appear randomly and unpredictably, but still following statistical patterns. The work done with this award will contribute to the understanding of these statistical patterns. This phenomenon is similar to turbulent fluid flow, which is also characterized by unpredictability that occurs with a statistical pattern. The award will also support the training of students in this exciting and broad field.
