# Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Theorem**

**1.**

**Remark**

**1.**

**Case 1**. $({U}_{0},{V}_{0},0)=\left(\right)open="("\; close=")">\frac{{\beta}_{1}+{\alpha}_{2}}{{\beta}_{1}+{\kappa}_{2}{\alpha}_{2}^{2}},\frac{{\beta}_{1}(1-{\kappa}_{2}{\alpha}_{2})}{{\beta}_{1}+{\kappa}_{2}{\alpha}_{2}^{2}},0$ (as $\omega \to -\infty $) and $(0,0,1)$ (as $\omega \to +\infty $).

**Case 2**. $({U}_{0},{V}_{0},0)$ (as $\omega \to -\infty $) and $(0,0,0)$ (as $\omega \to +\infty $).

**Case 3**. $(1,1,0)$ (as $\omega \to -\infty $) and $(0,1,0)$ (as $\omega \to +\infty $). This case occurs provided the additional restriction ${\alpha}_{2}=0$ takes place.

**Case 1**and use the tanh method. To the best of our knowledge paper [15] is one of the earliest works devoted to the tanh method (there are a lot recent papers, see, e.g., [17,27] and papers cited therein). However, it can be noted that there are not many papers devoted to application of this method to nonlinear systems of PDEs. The method is essentially based at the ad hoc ansatz [15]

**Case 2**, taking into account the corresponding steady-state points, we are looking for the traveling fronts in the form

**Case 3**, the exact solutions of system (6) were prescribed to have the form

**Remark**

**2.**

**Case 3**)

**Remark**

**3.**

## 3. Interpretation of Traveling Fronts

**(i)**$\mu >0$ and

**(ii)**$\mu <0$.

**(i)**, one immediately obtains $0<\mu <\frac{5}{2}$ (see the formula for ${\kappa}_{1}$ in (12)). For a simplicity, we assume additionally ${\alpha}_{2}={\alpha}_{4}\equiv \alpha $ and introduce the notations

**Remark**

**4.**

**(ii)**is essentially simpler. In fact, one immediately obtains ${\alpha}_{1}>0$ and ${\kappa}_{1}>0$ in (12). Assuming additionally that ${\alpha}_{2}=24{d}_{1}$ and solving the inequalities ${\beta}_{1}>0$ and ${\beta}_{3}>0$ (see (12)), we obtain the restrictions

**(i)**$\mu >0$ (Figure 1 and Figure 2) and Case

**(ii)**$\mu <0$ (Figure 3). All the curves satisfy the natural requirement of positivity at the given space intervals.

**(i)**. If we assume that the blue and green curves represent the communities of Russian language speakers and Ukrainian language speakers, while the red curve describes the frequency of bilingual speakers, then the real language shift occurred in Ukraine during the Soviet period (from the end of the Second WW till the USSR collapse) is qualitatively described by these curves. In fact, the language situation in Ukraine can be approximated by the 1D model because the communities of different language speakers varies very essentially from east to west (not so much from north to south).

**(ii)**$\mu <0$, so that the traveling fronts are moving to the left. As a result, the relevant interpretation is different. In fact, the time evolution leads to extinction of two communities, while only one monolingual community is the winner of this language competition.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Traveling fronts (11). Curves represent the functions $u({t}_{0},x)$ (blue represents the Russian speakers), $v({t}_{0},x)$ (red represents the bilingual speakers) and $w({t}_{0},x)$ (green represents the Ukrainian speakers) for the fixed time ${t}_{0}=0.01$ (

**left**) and ${t}_{0}=4$ (

**right**) and the parameters $\mu =\frac{3}{2},\phantom{\rule{4pt}{0ex}}{d}_{1}={d}_{3}=2,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}={\alpha}_{4}=5$ (other parameters are calculated by formulae (12)).

**Figure 2.**Traveling fronts (11). Curves represent the functions $u({t}_{0},x)$ (blue), $v({t}_{0},x)$ (red) and $w({t}_{0},x)$ (green) for the fixed time ${t}_{0}=6$ and the parameters $\mu =\frac{3}{2},\phantom{\rule{4pt}{0ex}}{d}_{1}={d}_{3}=2,\phantom{\rule{4pt}{0ex}}{\alpha}_{2}={\alpha}_{4}=5$ (other parameters are calculated by formulae (12)).

**Figure 3.**Traveling fronts (11). Curves represent the functions $u({t}_{0},x)$ (blue), $v({t}_{0},x)$ (red) and $w({t}_{0},x)$ (green) for the fixed time ${t}_{0}=0.01$ (

**left**) and ${t}_{0}=3$ (

**right**) and the parameters $\mu =-5,\phantom{\rule{4pt}{0ex}}{d}_{1}={d}_{3}=\frac{1}{2},\phantom{\rule{4pt}{0ex}}{\alpha}_{2}={\alpha}_{4}=12$ (other parameters are calculated by formulae (12)).

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**MDPI and ACS Style**

Cherniha, R.; Davydovych, V.
Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers. *Entropy* **2020**, *22*, 154.
https://doi.org/10.3390/e22020154

**AMA Style**

Cherniha R, Davydovych V.
Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers. *Entropy*. 2020; 22(2):154.
https://doi.org/10.3390/e22020154

**Chicago/Turabian Style**

Cherniha, Roman, and Vasyl’ Davydovych.
2020. "Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers" *Entropy* 22, no. 2: 154.
https://doi.org/10.3390/e22020154