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

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## Abstract

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## 1. Historical Remarks

**E**, applied to a semiconductor, the conductivity of the sample depends on

**E**, and the current–voltage curve becomes nonlinear. In some cases, calculations based on the assumption of the spatially homogeneous distribution of the current density,

**j**, and

**E**along and across the sample give rise to very unusual behavior of this dependence so that, in a certain area of the

**E**values, an increase in

**E**results in a decrease in

**j**.

**j**(

**E**) $\to j\left(E\right)$. Then, by definition, the conductivity $\sigma =j/E$. Let us define the differential conductivity as ${\sigma}_{\mathrm{d}}=\mathrm{d}j/\mathrm{d}E$. Thus, the area mentioned above is characterized with a negative differential conductivity. Here, I will not discuss the microscopic mechanisms explaining the negativeness of ${\sigma}_{d}$; a detailed description may be found, e.g., in Ref. [19]. Further increase in E makes ${\sigma}_{d}$ positive again so that the overall shape of the current–voltage curve resembles letter "N”, see Figure 1.

## 2. Problem Formulation

## 3. Stability Analysis

## 4. Spectrum of Schrödinger Equation

## 5. Conclusions

- The analysis of the linear stability of traveling wave solutions in a wide class of nonlinear diffusion problems is reduced to inspection of a bottom part of the spectrum of the associated Schrödinger equation, whose potential is generated by the profile of the analyzed solution.
- The translational invariance transformation generates in the stability spectrum a neutrally-stable (Goldstone) mode.
- The qualitative answer to the question about the stability of the solution is readily obtained based on the oscillation theorem—if the Goldstone mode does not have any nodes, the solution is stable. Otherwise, it is unstable.
- To quantitatively characterize the instability (if any), the “energy” level of the ground state of the Schrödinger equation should be obtained.
- A powerful tool to make the problem of a bottom part of the Schrödinger equation spectrum tractable is to approximate the potential by square wells.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A letter-N-shape current–voltage characteristics obtained under the assumption that electric field $E=const$ along and across the sample: the differential conductivity, ${\sigma}_{\mathrm{d}}$ is negative at ${E}_{c1}<E<{E}_{c2}$.

**Figure 2.**The only ever published piece of paper [1] (in Russian).

**Figure 3.**Phase plane $(\mathcal{E},{\mathcal{E}}_{\xi})$ (schematically). Three singular points are marked with red. The blue curve designates the homoclinic path corresponding to a single traveling domain. If the point $({\mathcal{E}}_{\mathrm{m}},0)$ merges with $({\mathcal{E}}_{3},0)$, the homoclinic path is split into two independent heteroclinic ones (the upper and lower parts of the homoclinic path, respectively). See text for details.

**Figure 4.**Schematically: The actual double-well potential $V\left(\xi \right)$ (smooth blue line). The approximation of $V\left(\xi \right)$ by the double-well square potential, ${V}_{\mathrm{DWS}}\left(\xi \right)$, is shown in black; ${\xi}_{1,2,3,4}$ designate the coordinates of the walls of the wells. ${U}_{1,2}\left(\xi \right)={V}_{\mathrm{DWS}}\left(\xi \right)+{V}_{1,2}\left(\xi \right)$ are the potentials of the single-well approximation, when tunneling is neglected.

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**MDPI and ACS Style**

Tribelsky, M.I.
Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential. *Physics* **2021**, *3*, 715-727.
https://doi.org/10.3390/physics3030043

**AMA Style**

Tribelsky MI.
Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential. *Physics*. 2021; 3(3):715-727.
https://doi.org/10.3390/physics3030043

**Chicago/Turabian Style**

Tribelsky, Michael I.
2021. "Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential" *Physics* 3, no. 3: 715-727.
https://doi.org/10.3390/physics3030043