# On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

## Abstract

**:**

## 1. Introduction

## 2. Typical Examples

## 3. Symmetries and Conservation Laws

#### 3.1. Symmetries

- Cylindrical Burgers is generated by $X,Y,Z$;
- Cylindrical KdV–Burgers is generated by $X,Z$;
- Spherical Burgers is generated by $X,Y,W$;
- Spherical KdV–Burgers is generated by $X,W$.

#### 3.2. Conservation Laws

- $n=0\Rightarrow $$\underset{T\to \infty}{lim}\frac{1}{T}\underset{0}{\overset{T}{\int}}{A}^{2}{sin}^{2}\left(\omega t\right)\phantom{\rule{0.166667em}{0ex}}dt==\frac{{A}^{2}}{2};$
- $n=\frac{1}{2}\Rightarrow $$\underset{T\to \infty}{lim}\frac{1}{T}\underset{0}{\overset{T}{\int}}\frac{1}{{T}^{\frac{1}{2}}}{t}^{\frac{1}{2}}(A{sin}^{2}\left(\omega t\right)\phantom{\rule{0.166667em}{0ex}}dt=\frac{{A}^{2}}{3};$
- $n=1\Rightarrow $$\underset{T\to \infty}{lim}\frac{1}{T}\underset{0}{\overset{T}{\int}}\frac{1}{T}t({A}^{2}{sin}^{2}\left(\omega t\right)\phantom{\rule{0.166667em}{0ex}}dt=\frac{{A}^{2}}{4}.$

## 4. Self-Similar Approximations To Solutions

## 5. Median Approximation

- For the cylindrical waves take$${\tilde{u}}_{2}=\frac{1}{2}[1-tanh\left(\frac{V}{{\epsilon}^{2}}(x-Vt)\right)]\xb7\frac{1}{3}\left(2V+V\sqrt{4-\frac{3x}{Vt}}\right);$$
- For spherical waves,$${\tilde{u}}_{3}=\frac{1}{2}[1-tanh\left(\frac{V}{{\epsilon}^{2}}(x-Vt)\right)]\xb7V\sqrt{e}exp\left(\mathrm{LambertW}\left(-\frac{x}{2Vt\sqrt{e}}\right)\right).$$

- For the cylindrical equation$${\int}_{0}^{Vt}\left[\frac{[1-tanh\left(\frac{V}{{\epsilon}^{2}}(x-Vt)\right)]}{2}\frac{1}{3}\left(2V+V\sqrt{4-\frac{3x}{Vt}}\right)\right]dx=\frac{32}{27}{V}^{2}t;$$
- For the spherical equation$${\int}_{0}^{Vt}\left[\frac{[1-tanh\left(\frac{V}{{\epsilon}^{2}}(x-Vt)\right)]}{2}V\sqrt{e}exp\left(\mathrm{LambertW}\left(\frac{-x}{2Vt\sqrt{e}}\right)\right)\right]dx=\frac{{V}^{2}t\xb7e}{2}.$$

## 6. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

KdV | Korteweg–de Vries |

IVBP | Initial value|boundary problem |

TWS | Traveling wave solution |

## References

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**Figure 1.**Cylindrical Burgers. $\epsilon =0.1,$

**Left**: ${u}_{0}=sint,\phantom{\rule{0.166667em}{0ex}}t=150.$

**Right**: ${u}_{0}=sin10t,\phantom{\rule{0.166667em}{0ex}}t=200.$

**Figure 2.**Spherical Burgers , ${u}_{0}=sint$.

**Left**: $\epsilon =0.1,\phantom{\rule{0.166667em}{0ex}}t=150$.

**Right**: ${\epsilon}^{2}=0.3,\phantom{\rule{0.166667em}{0ex}}t=150$.

**Figure 3.**Cylindrical KdV–Burgars.

**Left**: ${u}_{0}=sint,\phantom{\rule{0.166667em}{0ex}}t=300,\phantom{\rule{0.166667em}{0ex}}\epsilon =0.1,\phantom{\rule{0.166667em}{0ex}}\delta =0.001$.

**Right**: ${u}_{0}=3sint,\phantom{\rule{0.166667em}{0ex}}t=100,\phantom{\rule{0.166667em}{0ex}}\epsilon =0.1,\phantom{\rule{0.166667em}{0ex}}\delta \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.001.$

**Figure 4.**Spherical KdV–Burgers, ${u}_{0}=sint$.

**Left**: $t=300,\phantom{\rule{0.166667em}{0ex}}\epsilon =0.1,\delta =0.001.$

**Right**: $u\leftrightarrow -u,\phantom{\rule{0.166667em}{0ex}}t=300,\phantom{\rule{0.166667em}{0ex}}{\epsilon}^{2}=0.02,\phantom{\rule{0.166667em}{0ex}}\delta =0.001$ ${\epsilon}^{2}=0.2$

**Figure 5.**Constant boundary solutions to the Burgers equation, $\epsilon =0.1,t=200.$

**Left**: solid line—cylindrical; dotted line—spherical.

**Right**: A trace of movement to the right of the spherical solution at moments $t=37.5\xb7k,k=1\dots 6$.

**Figure 6.**

**Left**: solid line—solution to Equation (16); dotted line—its ${u}_{3}$ approximation.

**Right**: solid line—solution to spherical KdV, $x\to -x,\phantom{\rule{0.277778em}{0ex}}{\epsilon}^{2}=0.02,\phantom{\rule{0.277778em}{0ex}}\delta =0.002$; dotted line—its ${\tilde{u}}_{3}$ approximation; both at $t=200$.

**Figure 7.**Solid line—solution to spherical KdV, $x\to -x,\phantom{\rule{0.277778em}{0ex}}{\epsilon}^{2}=0.02,\phantom{\rule{0.277778em}{0ex}}\delta =0.002$, dotted line—its ${\tilde{u}}_{3}$ approximation; both at $t=400$.

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**MDPI and ACS Style**

Samokhin, A. On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations. *Symmetry* **2021**, *13*, 220.
https://doi.org/10.3390/sym13020220

**AMA Style**

Samokhin A. On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations. *Symmetry*. 2021; 13(2):220.
https://doi.org/10.3390/sym13020220

**Chicago/Turabian Style**

Samokhin, Alexey. 2021. "On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations" *Symmetry* 13, no. 2: 220.
https://doi.org/10.3390/sym13020220