Colloquium

Speaker: Pavel Lushnikov (University of New Mexico)

Abstract: Many nonlinear systems of partial differential equations have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which is called by collapse. Near singularity point there is a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability and other mechanisms become important such as inelastic collisions in the Bose-Einstein condensate, optical breakdown and dissipation in nonlinear optical media and plasma, wave breaking in hydrodynamics. Collapses occur in numerous reduced physical and biological systems including a nonlinear Schrodinger quation (NLSE) and a Keller-Segel equation (KSE). We will focus on the collapse in the critical spatial dimension two (2D) which has numerous applications. For instance, 2D NLSE describes the propagation of the intense laser beam in nonlinear Kerr media (like usual glass) which results in the catastrophic self-focusing (collapse) eventually causing optical damage as was routinely observe in experiment since 1960-es. Recently such events have been also often referred as optical rogue waves. Another dramatic NLSE application is the formation of rogue waves in ocean. 2D KSE collapse describes the bacterial aggregation in Petri dish as well as the gravitational collapse of Brownian particles. We study the universal self-similar scaling near collapse, i.e. the spatial and temporal structures near blow up point. In the critical 2D case all these collapses share a strikingly common feature that the collapsing solutions have a form of either rescaled soliton (for NLSE) or rescaled stationary solution (for KSE). The time dependence of that scale determines the time-dependent collapse width L(t) and amplitude ~1/L(t). At leading order L(t)~ (t_c-t)^{1/2} for all mentioned equations, where t_c is the collapse time. Collapse however requires the modification of that scaling which in NLSE has the well-known loglog type ~ (\ln|\ln(t_c-t)|)^{-1/2} as well as KSE has another well-known type of logarithmic scaling modification. Loglog scaling for NLSE was first obtained asymptotically in 1980-es and later proven in 2006. However, it remained a puzzle that this scaling was never clearly observed in simulations or experiment. Similar situation existed for KSE. Here solved that puzzle by developing a perturbation theory beyond the leading order logarithmic corrections for both NLSE and KSE. We found that the classical loglog modification NLSE requires double-exponentially large amplitudes of the solution ~10^10^100, which is unrealistic to achieve in either physical experiments or numerical simulations. In contrast, we found that our new theory is valid starting from quite moderate (about 3 fold) increase of the solution amplitude compare with the initial conditions. We obtained similar results for KSE. In both cases new scalings are in excellent agreement with simulations. This efficiency of analytical results also allowed to study 2D NLSE-type dissipative system in the conditions of multiple random spontaneous formation of collapses in space and time. Dissipation ensures collapse regularization while collapses are responsible for non-Gaussian tails in the probability density function of amplitude fluctuations which makes turbulence strong. Power law of non-Gaussian tails is obtained for strong NLSE turbulence which is a characteristic feature of rogue waves. We suggest the spontaneous formation optical rogue from turbulent as a perspective route to the combing of multiple laser beams, generated by a number of fiber lasers, into a single coherent powerful laser beam.

Short bio: Pavel Lushnikov received PhD in theoretical physics from the Landau Institute of Theoretical Physics. He moved to the postoctoral postion at the Los Alamos National Laboratory and later became the assistant professor at the University of Notre Dame. He joined UNM Department of Mathematics and Statistics in 2006 as the associate professor and has been serving as the full professor since 2012. Since 2006 he has been awarded by 5 NSF and NSF/DOE grants. In 2008 he received Doctor of Science Degree in Physical and Mathematical Sciences, highest scientific degree in Russia, awarded for major scientific achievements beyond PhD by the Landau Institute. His interests include a wide range of topics in applied mathematics, nonlinear waves and theoretical physics. Among them are laser fusion and laser-plasma interaction; dynamics of fluids with free surface and nonlinear interactions of surface waves; theory of the wave collapse, singularity formation and its application to plasma physics, hydrodynamics, biology and nonlinear optics; high-bit-rate optical communication; dispersion-managed optical fiber systems; solution propagation in optical systems; high performance parallel simulations of optical fiber systems; Bose-Einstein condensation of ultracold dipolar gases. His recent Optics Letters paper on nonlinear beam combining, published in June 2014, has been in the top download of Optics Letters since then.