# Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation

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## Abstract

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## 1. Introduction

## 2. Description of the Auxiliary Equation Method

**Step 1.**- To find the exact traveling waves of Equation (7), we introduce the wave variable$$u(x,t)=U\left(\xi \right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\xi =x-\omega t,$$$$G(U,{U}_{\xi},{U}_{\xi \xi},\dots )=0.$$
**Step 2.**- We assume that Equation (9) has the finite series form solution$$U\left(\xi \right)=\sum _{k=0}^{K}{\gamma}_{k}\phantom{\rule{0.166667em}{0ex}}{\varphi}^{k}\left(\xi \right),$$
**Step 3.**- The integer K can be computed by balancing the nonlinear terms and the highest order derivatives arising in Equation (9). We denote the degree of $U\left(\xi \right)$ by $Deg\left(U\right(\xi \left)\right)=K$ which leads to the degrees of other expressions as$$Deg\left(\right)open="("\; close=")">\frac{{d}^{r}U}{d{\xi}^{r}}$$
**Step 4.**- By substituting Equation (6) with Equation (10) into Equation (9), we get an algebraic equation involving powers of $\varphi \left(\xi \right)$. Then, equating the coefficients of each power of $\varphi \left(\xi \right)$ to zero yields a system of algebraic equations for ${\mu}_{2},$${\mu}_{3},$${\mu}_{4},$$\omega $ and ${\gamma}_{k}$$(k=0,1,2,\dots ,K)$.
**Step 5.**- After solving the set of over-determined algebraic equations with the aid of Maple, one ends up with the explicit expressions for ${\mu}_{2},$${\mu}_{3},$${\mu}_{4},$$\omega $ and ${\gamma}_{k}$$(k=0,1,2,\dots ,K)$.
**Step 6.**- Consequently, we may obtain different types of exact traveling wave solutions for Equation (7), such as solitons; kink and anti-kink, bell and anti-bell, periodic, and exponential solutions; and other solutions by substituting ${\mu}_{2}$, ${\mu}_{3}$, ${\mu}_{4}$, $\omega $ and ${\gamma}_{k}$$(k=0,1,2,\dots ,K)$ and the general solutions of Equation (6) into Equation (10).

**i.**- When ${\mu}_{2}>0,$$$\begin{array}{ccc}\hfill {\varphi}_{1}\left(\xi \right)& =& \frac{-{\mu}_{2}{\mu}_{3}{sech}^{2}\left(\right)open="("\; close=")">\frac{\sqrt{{\mu}_{2}}}{2}\xi}{}{\mu}_{3}^{2}-{\mu}_{2}{\mu}_{4}{\left(\right)}^{1}2\hfill \\ ,\end{array}$$
**ii.**- When ${\mu}_{2}>0$ and $\Delta >0,$$${\varphi}_{4}\left(\xi \right)=\frac{2{\mu}_{2}sech\left(\right)open="("\; close=")">\sqrt{{\mu}_{2}}\xi}{}\epsilon \sqrt{\Delta}-{\mu}_{3}sech\left(\right)open="("\; close=")">\sqrt{{\mu}_{2}}\xi .$$
**iii.**- When ${\mu}_{2}>0$ and $\Delta =0,$$$\begin{array}{ccc}\hfill {\varphi}_{5}\left(\xi \right)& =& -\frac{{\mu}_{2}}{{\mu}_{3}}\left(\right)open="("\; close=")">1+\epsilon tanh\left(\right)open="("\; close=")">\frac{\sqrt{{\mu}_{2}}}{2}\xi \hfill & ,\end{array}\hfill {\varphi}_{6}\left(\xi \right)& =& -\frac{{\mu}_{2}}{{\mu}_{3}}\left(\right)open="("\; close=")">1+\epsilon coth\left(\right)open="("\; close=")">\frac{\sqrt{{\mu}_{2}}}{2}\xi \hfill & ,$$
**iv.**- When ${\mu}_{2}>0$ and $\Delta <0,$$${\varphi}_{8}\left(\xi \right)=\frac{2{\mu}_{2}csch\left(\right)open="("\; close=")">\sqrt{{\mu}_{2}}\xi}{}\epsilon \sqrt{-\Delta}-{\mu}_{3}csch\left(\right)open="("\; close=")">\sqrt{{\mu}_{2}}\xi .$$
**v.**- When ${\mu}_{2}<0$ and $\Delta >0,$$${\varphi}_{9}\left(\xi \right)=\frac{2{\mu}_{2}sec\left(\right)open="("\; close=")">\sqrt{-{\mu}_{2}}\xi}{}\epsilon \sqrt{\Delta}-{\mu}_{3}sec\left(\right)open="("\; close=")">\sqrt{-{\mu}_{2}}\xi .$$

## 3. Application of the Auxiliary Equation Method

#### 3.1. Traveling Wave Solutions for the Case $\delta \in \mathbb{R}$

#### 3.2. Traveling Wave Solutions for the Case $\delta \in \mathbb{R}\backslash \{\pm 1\}$

#### 3.3. Traveling Wave Solutions for the Case $\delta \in \mathbb{R}\backslash \{-1\}$

#### 3.4. Traveling Wave Solutions for the Case $\delta =1$

#### 3.5. Traveling Wave Solutions for the Case $\delta =-1$

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The exact solutions of the generalized SIdV Equation (5) over the time interval $[0,10]$. The

**left**is a bell-shaped solitary wave solution (18) when ${\gamma}_{1}=1,\delta =2,{\mu}_{2}=1,{\mu}_{3}=-1$, and $\epsilon =1$, the

**middel**is an anti-kink-shaped solitary wave solution (39) when $\delta =\frac{1}{2},{\mu}_{2}=\frac{1}{4},{\mu}_{3}=1,{\mu}_{4}=1$, and $\epsilon =-1$, and the

**right**is a periodic solitary wave solution (43) when $\delta =2,{\mu}_{2}=\frac{-1}{5},{\mu}_{3}=1,{\mu}_{4}=-1$, and $\epsilon =1$.

**Figure 2.**The exact solutions of the generalized SIdV Equation (5) at $t=0$. The

**left**is a singular wave solution (18) when ${\gamma}_{1}=1,\delta =2,{\mu}_{2}=1,{\mu}_{3}=1$, and $\epsilon =1$, the

**middel**is a periodic singular wave solution (19) when $\delta =2,{\mu}_{2}=-1,{\mu}_{3}=1,{\gamma}_{1}=1$, and $\epsilon =1$, and the

**right**is a kink-shaped solitary wave solution (48) when $\delta =1,{\mu}_{3}=1,{\gamma}_{0}=1,{\gamma}_{1}=1,$ and $\epsilon =-1$.

**Figure 3.**The exact solutions of the generalized SIdV Equation (5) at $t=0$. The

**left**is a bell-shaped solitary wave solution (51); namely, $u(x,t)={sech}^{4}(\frac{1}{2}x-\frac{4}{3}t)$ when $\delta =1,\gamma =\frac{4}{3},{\gamma}_{2}=1,{\mu}_{2}=1$, and ${\mu}_{3}=1$, the

**middle**is a bell-shaped solitary wave solution (59); namely, $u(x,t)=sech(x-\frac{4}{3}t)$ when $\delta =1,\gamma =\frac{2}{3},{\gamma}_{1}=1,{\mu}_{2}=1,{\mu}_{4}=-1$, and $\epsilon =1$, and the

**right**is a bell-shaped solitary wave solution (60); namely, $u(x,t)=\frac{1}{2}+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{sech}^{2}(\frac{1}{2}x-2t)$ when $\delta =-1,{\gamma}_{0}=\frac{1}{2},{\gamma}_{1}=-1,{\mu}_{2}=1$, and ${\mu}_{3}=2$.

**Figure 4.**The exact peakon wave solutions of the generalized SIdV Equation (5) over the time interval $[0,5]$. The

**left**side shows a peakon wave solution (44)—namely, $u(x,t)=\frac{1}{2}\left(\right)open="["\; close="]">1-tanh\left(\frac{1}{2}\right|x-t\left|\right)$ when $\delta =\frac{1}{2}$ and ${\mu}_{2}=1$; and the

**right**side shows a peakon wave solution (55), namely, $u(x,t)=exp(-|x-t\left|\right)$ when $\delta =1,\gamma =1,{\mu}_{2}=1,$ and ${\gamma}_{1}=\frac{1}{4}$. The peakon solution is obtained when kink and anti-kink solutions (or two exponential solutions of the opposite forms) move in the same direction with the same speed.

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**MDPI and ACS Style**

Alzaleq, L.; Manoranjan, V.; Alzalg, B.
Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation. *Mathematics* **2022**, *10*, 414.
https://doi.org/10.3390/math10030414

**AMA Style**

Alzaleq L, Manoranjan V, Alzalg B.
Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation. *Mathematics*. 2022; 10(3):414.
https://doi.org/10.3390/math10030414

**Chicago/Turabian Style**

Alzaleq, Lewa’, Valipuram Manoranjan, and Baha Alzalg.
2022. "Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation" *Mathematics* 10, no. 3: 414.
https://doi.org/10.3390/math10030414