Nonlinear Waves Seminar - Pavel Lushnikov
Pavel Lushnikov, Department of Mathematics, University of New Mexico
Tail minimization principle in wave collapse
Wave collapse occurs in numerous nonlinear physical systems resulting in catastrophic self-focusing of laser beams in optical media, collapse of Bose-Einstein condensate and white foam formation on the crests of oceanic waves. Underlying equations for all these diverse effects are two-dimensional nonlinear Schrodinger equation (NLSE) and its nonlocal extension, Davey-Stewartson equation (DSE). We find that collapses in both NLSE and DSE obey the tail minimization principle when physical systems dynamically choose self-similar-type solutions which minimize the spatial tails of the collapsing solution on the border of the spatial collapsing region. Qualitatively similar minimization occurs in the collapse in Keller-Segel equation of bacterial aggregation. This minimization ensures that the singularity (collapse) reaches in fastest possible time (propagation distance for optical applications) because the maximum optical power (in optical applications) or number of particles (in Bose-Einstein condensate) are captured in the collapsing region. A weak escape of particles (optical power) from that region is controlled by an analog of quantum tunneling resulting both for NLSE and DSE in square root time dependence of spatial scaling with double logarithmic modification. We found that such scaling is asymptotically dominant only after the amplitude of collapsing solution reaches 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 developed a detailed an asymptotic theory which is valid starting from quite moderate (about 3 fold) increase of the solution amplitude compared with the initial conditions.
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Tuesday, January 25, 2022 at 4:00pm to 5:00pm
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