Prediction of random and chaotic dynamics in nonlinear optics
The prediction of interactions between multiple high-power laser beams is a longstanding open problem in physics. One traditional assumption in this area is that these interactions are deterministic. We show, however, that a single beam may lose its initial phase information in the presence of input noise. This "loss of phase", obtained in the classical description of nonlinear laser propagation, the Nonlinear Schrodinger equation (NLS), renders the multiple-beams interactions unpredictable.
Computationally, the study of the NLS or other differential equations with random inputs is made possible by a novel algorithm which efficiently estimates probability density functions (PDF) that arise in these settings. This task leads to a general question in numerical uncertainty-quantification (UQ): given a function and its approximation, are the two PDFs they induce similar? The analysis of this question leads through an interesting path, which lies at the intersections of probability, approximation, and optimal transport theory.