The focusing nonlinear Schrödinger equation arises in various physical phenomena and it is therefore of interest to determine mathematical conditions on the initial data that guarantee whether the corresponding solution will blow up in finite time or exist globally in time. We focus on solutions to the mass‐supercritical nonlinear Schrödinger equation (1) in 1D case. In particular, we investigate numerical thresholds between blow up and scattering of solutions in the focusing NLS equation as well as rates and profiles of blow up solutions. We consider several initial data families and the degree of nonlinearity p>5.
The role of fluid motion in delivery of nutrients to phytoplankton cells is a fundamental question in biological and chemical oceanography. In the study of mass transfer to phytoplankton, diatoms are of particular interest. They are non-motile, are often the most abundant components in aggregates and often form chains, so they are the ones expected to benefit most from enhancement of nutrient flux due to dissipating turbulence. Experimental data to test the contribution of advection to nutrient acquisition by phytoplankton are scarce, mainly because of the inability to visualize, record and thus imitate fluid motions in the vicinities of cells in natural flows. Laboratory experiments have most often used steady Couette flows to simulate the effects of turbulence on plankton. However, steady flow, producing spatially uniform shear, fails to capture the diffusion of momentum and vorticity, the essence of turbulence. Thus, numerical modelling plays an important role in the study of effects of fluid motion on diffusive and advective nutrient fluxes. In this paper we use the immersed boundary method to model the interaction of rigid and flexible diatom chains with the surrounding fluid and nutrients. We examine this interaction in two nutrient regimes, a uniform background concentration of nutrients, such as might be typical of an early spring bloom, and a contrasting scenario in which nutrients are supplied as small, randomly distributed pulses, as is more likely for oligotrophic seas and summer conditions in coastal and boreal seas. We also vary the length and flexibility of chains, as whether chains are straight or bent, rigid or flexible will affect their behaviour in the flow and hence their nutrient fluxes. The results of numerical experiments suggest that stiff chains consume more nutrients than solitary cells. Stiff chains also experience larger nutrient fluxes compared to flexible chains, and the nutrient uptake per cell increases with increasing stiffness of the chain, suggesting a major advantage of silica frustules in diatoms.