17 research outputs found
Statistics and Characteristics of Spatio-Temporally Rare Intense Events in Complex Ginzburg-Landau Models
We study the statistics and characteristics of rare intense events in two
types of two dimensional Complex Ginzburg-Landau (CGL) equation based models.
Our numerical simulations show finite amplitude collapse-like solutions which
approach the infinite amplitude solutions of the nonlinear Schr\"{o}dinger
(NLS) equation in an appropriate parameter regime. We also determine the
probability distribution function (PDF) of the amplitude of the CGL solutions,
which is found to be approximately described by a stretched exponential
distribution, , where . This
non-Gaussian PDF is explained by the nonlinear characteristics of individual
bursts combined with the statistics of bursts. Our results suggest a general
picture in which an incoherent background of weakly interacting waves,
occasionally, `by chance', initiates intense, coherent, self-reinforcing,
highly nonlinear events.Comment: 7 pages, 9 figure
Beam stabilization in the two-dimensional nonlinear Schrodinger equation with an attractive potential by beam splitting and radiation
Renormalizing Partial Differential Equations
In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]