Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Shrödinger Equation with Complicated Potential

The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically the problem is reduced to the calculation of the "energy" of the ground state in Schrödinger equation with a complicated potential. A general method to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells is proposed and applied. Possible generalization of the approach to other types of nonlinear diffusion equations is discussed.

Geometric Effects of a Quarter of Corrugated Torus

In the spirit of the thin-layer quantization scheme, we give the effective Shrödinger equation for a particle confined to a corrugated torus, in which the geometric potential is substantially changed by corrugation. We find the attractive wells reconstructed by the corrugation not being at identical depths, which is strikingly different from that of a corrugated nanotube, especially in the inner side of the torus. By numerically calculating the transmission probability, we find that the resonant tunneling peaks and the transmission gaps are merged and broadened by the corrugation of the inner side of torus. These results show that the quarter corrugated torus can be used not only to connect two tubes with different radiuses in different directions, but also to filter the particles with particular incident energies.

MASS CONCENTRATION WINDOW SIZE AND STRICHARTZ NORM DIVERGENCE RATE FOR THE L2-CRITICAL NONLINEAR SHRÖDINGER EQUATION

We observe a link between the window size of mass concentration and the rate of explosion of the Strichartz norm by revisiting Bourgain's mass concentration for the L2-critical nonlinear Schrödinger equations.