Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger 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 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.

in the names meant the drastic difference in the ranks. Though BSU was quite a good 23 university, MSU was (and is) the Number One. 24 Doing paperwork related to the transfer, I asked Prof. El'yashevich for a reference 25 letter to somebody of his collaborators in Moscow. I got this letter to his former Ph.D. 26 student Dr. Sergei Ivanovich Anisimov [3], who became my next scientific adviser. 27 It was 1969. At that time, I could not even imagine how lucky I was. Dr. Anisimov   In what follows only one-dimensional cases will be considered, so I can replace 85 j(E) → j(E). Then, by definition, the conductivity σ = j/E. Let us define the differential 86 conductivity as σ d = dj/dE. Thus, the area mentioned above is characterized with a 87 1 There is an interesting story related to this abbreviation. When another Prof. Landau's disciple, Prof. Alexander Solomonovich Kompaneetz, known, in addition to his outstanding scientific results, for his sense of humor, leant about LAK-theory, he said, "It is excellent that the authors did not employ the inverted order of them." (kal in Russian means excrements.) negative differential conductivity. Here I will not discuss the microscopic mechanisms 88 explaining the negativeness of σ d ; their detailed description may be found, e.g., in 89 Ref. [9]. Further increase in E makes σ d positive again so that the overall shape of the 90 current-voltage curve resembles the letter "N". direction, the domain expands and transforms into two traveling layers [9]. excited states [12]. However, at that time, I was much more ignorant than I am now.    Fig. 1 is the only remained piece of the paper 2 .

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The next attempt to publish these results I made after the defense of my MS thesis.

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An appointed referee of the thesis was another disciple of Prof. Landau, Prof. Igor

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Ekhiel'evich Dzyaloshinskii [14].  it was pretty unexpected. The Editors informed me that they had never received my 206 manuscript.

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By that time, on the one hand, I had already published paper [15], where the 208 secondary instability of the Gunn domain was inspected just employing the Ritz method. The growth of instability of wide Gunn domains (the width of the top is much larger than the widths of the faces) at the stationary external current regime is inspected. The basis of calculations is the phenomenological model, in which the total current is composed of the conductivity current and the diffusion one. Instability affects the domain faces so that they begin to shift in opposite directions. The instability increment is calculated. The diffusion coefficient is supposed to be independent of the field. Mathematically the stability problem is reduced to a one-dimensional Schrödinger equation with a certain complicated potential. It is shown that the results are weakly dependent on details of this potential. Therefore, the potential is approximated by two square potential wells (separated by a barrier), which made it possible to obtain an explicit expression for the increment. The problem is solved in two limiting cases, namely symmetric and highly asymmetric domains. In both cases, additional drops of voltage on the domain are calculated, as well as the width of the faces and the top of the domain. It is essential that the final result does not include any integral characteristics of the problem. It depends only on the value of the functions j(E) and dj/dE = σ d at certain characteristic points. A developed method without significant changes may be extended to the study of the instability of waves in the form of two or more domains.
On the other hand, I got a job and, owing to that, was forced to abandon my study in available to the international community. Perhaps fifty years is a too long period to 214 complete a task, but "that is not lost that comes at last!"

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In the end of this, maybe longsome remarks, I have to say that the results discussed 216 below are not exactly the same as those in Ref. [1]. Firstly, it is not good to publish the

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Thus, the problem is to find the instability increment for a single traveling Gunn 227 domain at a fixed current in the circuit. According to what has been said above, the 228 current-voltage characteristic of the semiconductor sample in question has the shape 229 schematically shown in Fig. 2. Regarding the external current j ext , I suppose that it 230 satisfies the restrictions j c1 < j ext < j c2 so that the equation j(E) = j ext always has tree 231 roots E 1,2,3 .

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It is convenient to normalize the electric field over E 2 and j(E) over j ext introducing the dimensionless quantities E ≡ E/E 2 and u(E ) ≡ j(E)/j ext . Then, under certain assumptions, in the traveling coordinate frame connected with the domain, the normalized electric field in the sample is described by the following equation [9]: where the subscripts indicate the corresponding derivatives. Eq. (1) is written in dimen-233 sionless variables, whose detailed definition is not important for the subsequent analysis 234 (it may be found in Ref. [9]). Note only that D, s, ξ, and η stand for the diffusion 235 coefficient, the domain velocity in the laboratory coordinate frame, traveling coordinate,  We suppose that D = constant. This assumption simplifies calculations, but it is 239 not crucial for the analysis. A more general case, when D = D(E ), was inspected by 240 Knight and Peterson [11].

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It is important to stress that, if the dependence u(E ) is not related to the specific

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It is possible to show that such a solution of Eq. (1) exists at s = 1 solely [11]. Since 254 this is the only case I am interested in, s below is always supposed to be equal to unity. 255 Then, the homoclinic path may be found explicitly [11]; however, I do not need this

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The stability analysis performed in Ref. [11] generalizes a very beautiful approach 261 developed by Zeldovich and Barenblatt for the inspection of the stability of a slow 262 combustion front [16]. Its main idea is as follows. It is supposed that δE (ξ, η) = 263 E (1) (ξ) exp(−λη), where λ is an eigenvalue of the stability problem. If there is a negative 264 λ in the problem's spectrum, it means instability.

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However, I have departed from the stability analysis of the Gunn domain. It is high time to be back. Actually, not so much remains to do. Collecting together all mentioned above, one can conclude that is the eigenfunction of Shrödinger equation, Eqs. (4)-(6) with zero eigenvalue.

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Remember now the oscillation theorem [12]. It states that in a 1D Shrödinger equation

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Thus, to get the value of the instability increment and hence, the characteristic 327 time for the traveling domain decomposition, one has to obtain the energy level for the 328 ground state in the Shrödinger equation with the complicated potential given by Eq. (6).

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As it has been mentioned above, the main idea employed in Ref. [1] to fulfill this task 330 is to approximate the actual smooth potential by a superposition of square potential   Thus, I suppose that the wave functions of the ground states for each well |1, 2⟩ are known and that these wave functions satisfy the equationsĤ 1,2 |1, 2⟩ = 0. HereĤ 1,2 designate the Hamiltonians, whose potentials V 1,2 include the corresponding single well solely. Then, I look for the wave function of the complete problem with the two-well potential in a form of a linear superposition of |1, 2⟩: where c 1,2 are constants, which should be defined in the course of calculations. Next, since |ψ⟩ is an eigenfunction of the complete HamiltonianĤ Making scalar products with ⟨1, 2| and taking into account the normalization conditions 369 ⟨1|1⟩ = ⟨2|2⟩ = 1, one arrives from Eq. (9) to the following equations for c 1,2 : To calculate the matrix elements, it is convenient to single out from the full potential the part corresponding to a single well, i.e. to suppose that V(ξ) = U 1,2 − V 1,2 , see Fig. 4, where for the first well (left in Fig. 4) V 1 = 0 at ξ < ξ 3 , while at ξ > ξ 3 the sum of V and V 1 equals the hight of the barrier. Analogously, for the second well (right) the sum V + V 2 equals the hight of the barrier at ξ < ξ 2 , while at ξ > ξ 2 the potential V 2 = 0. Thus, U 1,2 are the single-well potentials, and the corresponding Hamiltonians acting on the reciprocal wave functions produce zero. Then, where d b = ξ 3 − ξ 2 is the barrier width. The same estimate is true for H 22 , where d 1,2 stand for the widths of the corresponding wells. In the same manner one obtains Then, in the leading approximation Thus, as it could be expected, the value of the instability increment for the broad domain 381 is exponentially small indeed.

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Summarizing and generalizing the discussed above, one may arrive at the following