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

Michael I Tribelsky

doi:10.3390/physics3030043

Abstract

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 the 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.

Highlights

Historical RemarksWhen the Editors kindly offered me to submit a paper to this Special Issue dedicated to my fifty years in physics, I began to think about a possible topic of the paper
The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors
The point is that though I graduated from the Lomonosov Moscow State University (MSU)—the one where I head a laboratory—I did not enter this university in the usual, standard manner

Summary

Historical Remarks

When the Editors kindly offered me to submit a paper to this Special Issue dedicated to my fifty years in physics, I began to think about a possible topic of the paper. He moved to Moscow from Khar’kov (a big Ukrainian city), where he resided before In addition to this position, Il’ya Mikhailovich got a professorship at the Chair for Quantum Theory, the Faculty of Physics, MSU. On the one hand, I had already published a paper [23], where the secondary instability of the Gunn domain was inspected just employing the Ritz method. I extracted from this old problem the essential points and generalized them These points are as follows: (i) the conclusion about the instability of traveling pulses in a broad class of nonlinear diffusion-type problems and (ii) a new method to obtain the bottom-part spectrum of the Schrödinger equation with a complicated potential.

Problem Formulation

Stability Analysis

Spectrum of Schrödinger Equation

Conclusions