# Integrability Properties of the Slepyan–Palmov Model Arising in the Slepyan–Palmov Medium

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## Abstract

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## 1. Introduction

- $\rho $ represents the mass density of the carrier medium.
- $\varrho $ and $\mu $ are the Lame elastic moduli characterizing the carrier medium’s elasticity.
- $\mathbf{v}$ denotes the displacement vector of points within the carrier medium.
- $\mathbf{K}$ stands for the intensity of the external body force acting on the medium.
- ${\mathbf{w}}_{q}$ represents the absolute displacement vector of the oscillator mass with respect to its equilibrium position.
- ${\mathbf{Q}}_{q}$ denotes the external force applied to the mass of the oscillator.
- The quantity $m\left(q\right)\mathrm{d}q$ corresponds to the mass of all oscillators with eigenfrequencies lying within the interval $(q,q+\mathrm{d}q)$ multiplied by a unit volume.
- $m={\int}_{0}^{\infty}m\left(q\right)\mathrm{d}q$ is the total mass density of all oscillators fixed to the carrier medium.
- $c\left(q\right)={q}^{2}m\left(q\right)$ represents the static stiffness of the oscillator suspension.

## 2. Lie Group Method

**Proposition**

**1.**

**Proof.**

**Case****1:**- ${\mu}_{4}\ne 0,{\mu}_{3}=0,{\mu}_{1}=0$. Then we have,

**Case 2:**${\mu}_{4}\ne 0,{\mu}_{3}=0,{\mu}_{1}\ne 0$. Then we have,

**Case 3:**${\mu}_{4}\ne 0,{\mu}_{3}\ne 0,{\mu}_{1}=0$. Then we have,

**Case 4:**${\mu}_{4}\ne 0,{\mu}_{3}\ne 0,{\mu}_{1}\ne 0$. Then we have,

**Case 5:**${\mu}_{4}=0,{\mu}_{3}=0,{\mu}_{1}=0$. Then we have,

**Case 6:**${\mu}_{4}=0,{\mu}_{3}=0,{\mu}_{1}\ne 0$. Then we have,

**Case 7:**${\mu}_{4}=0,{\mu}_{3}\ne 0,{\mu}_{1}=0,{\mu}_{2}=0$. Then we have,

**Case 8:**${\mu}_{4}=0,{\mu}_{3}\ne 0,{\mu}_{1}=0,{\mu}_{2}\ne 0$. Then we have,

**Case 9:**${\mu}_{4}=0,{\mu}_{3}\ne 0,{\mu}_{1}\ne 0$. Then we have,

## 3. Similarity Reductions and Invariant Solutions

**Vector field ${\mathcal{J}}_{6}=\langle {\mathcal{E}}_{1}\rangle $**.

**Vector field ${\mathcal{J}}_{7}=\langle {\mathcal{E}}_{3}\rangle $**.

**Vector field ${\mathcal{J}}_{2}=\langle {\mathcal{E}}_{1}+c{\mathcal{E}}_{4}\rangle $**.

**Vector field ${\mathcal{J}}_{8}=\langle {\mathcal{E}}_{2}+c{\mathcal{E}}_{3}\rangle $**.

**Vector field ${\mathcal{J}}_{3}=\langle {\mathcal{E}}_{3}+c{\mathcal{E}}_{4}\rangle $**.

**Vector field ${\mathcal{J}}_{9}=\langle {\mathcal{E}}_{1}+c{\mathcal{E}}_{3}\rangle $**.

**Vector field ${\mathcal{J}}_{4}=\langle {\mathcal{E}}_{1}+c{\mathcal{E}}_{3}+l{\mathcal{E}}_{4}\rangle $**.

## 4. Local Conservation Laws via Homotopy Operator

**Proposition**

**2.**

- Using the characteristic ${\Psi}_{1}=\frac{1}{6}{t}^{3}+\frac{1}{2}{x}^{2}t$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{1}^{t}\hfill & =-\frac{1}{3}\beta {\theta}_{txx}{\theta}_{x}{x}^{2}t-\beta {x}^{2}t{\theta}_{txx}+\beta {t}^{2}{\theta}_{xx}{\theta}_{x}+\beta {x}^{2}{\theta}_{x}{\theta}_{xx}-\frac{1}{3}\beta {t}^{2}{\theta}_{tx}{\theta}_{xx}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & -\beta {x}^{2}t{\theta}_{tx}{\theta}_{xx}-\frac{1}{2}\theta {x}^{2}+\frac{1}{6}{t}^{3}{\theta}_{t}+\frac{1}{2}{x}^{2}t{\theta}_{t}+\frac{1}{2}{t}^{2}{\theta}_{xx}+\frac{1}{2}{x}^{2}{\theta}_{xx}-\frac{1}{6}{t}^{3}{\theta}_{txx}\hfill \\ & -\frac{1}{2}{x}^{2}t{\theta}_{txx},\hfill \\ {\Omega}_{1}^{x}\hfill & ={\theta}_{xt}-{\alpha}^{2}t{\theta}_{x}-\beta t{\theta}_{x}^{2}-\frac{1}{2}{x}^{2}t{\theta}_{x}-\frac{1}{6}{\alpha}^{2}{t}^{3}{\theta}_{xxx}-t{\theta}_{x}-\frac{1}{6}{t}^{3}{\theta}_{x}\hfill \\ & +{\alpha}^{2}xt{\theta}_{xx}-\frac{1}{2}{\alpha}^{2}{x}^{2}t{\theta}_{xxx}.\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{2}=\frac{1}{2}{t}^{2}+\frac{1}{2}{x}^{2}$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{2}^{t}\hfill & =-\beta {t}^{2}{\theta}_{x}{\theta}_{txx}-\beta {x}^{2}{\theta}_{x}{\theta}_{txx}+2\beta t{\theta}_{x}{\theta}_{xx}-\beta {t}^{2}{\theta}_{tx}{\theta}_{xx}-\beta {x}^{2}{\theta}_{tx}{\theta}_{xx}-t\theta \hfill \\ & +\frac{1}{2}{t}^{2}{\theta}_{t}+\frac{1}{2}{x}^{2}{\theta}_{t}+t{\theta}_{xx}-\frac{1}{2}{t}^{2}{\theta}_{txx}-\frac{1}{2}{x}^{2}{\theta}_{txx},\hfill \\ {\Omega}_{2}^{x}\hfill & ={\alpha}^{2}x{\theta}_{xx}-\frac{1}{2}{\alpha}^{2}{t}^{2}{\theta}_{xxx}-\frac{1}{2}{\alpha}^{2}{x}^{2}{\theta}_{xxx}-\frac{1}{2}{t}^{2}{\theta}_{x}-\frac{1}{2}{x}^{2}{\theta}_{x}-\beta {\theta}_{x}^{2}\hfill \\ & -x\theta -{\theta}_{x}-{\alpha}^{2}{\theta}_{x}.\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{3}=-{\alpha}^{2}tcos\left(\frac{x}{\alpha}\right)$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{3}^{t}\hfill & ={\alpha}^{2}cos\left(\frac{x}{\alpha}\right)(2\beta t{\theta}_{tx}{\theta}_{xx}+2\beta t{\theta}_{x}{\theta}_{txx}-2\beta {\theta}_{x}{\theta}_{xx}-t{\theta}_{t}+t{\theta}_{txxx}+\theta -{\theta}_{xx}),\hfill \\ {\Omega}_{3}^{x}\hfill & ={\alpha}^{3}t(\alpha {\theta}_{xx}cos\left(\frac{x}{\alpha}\right)+{\theta}_{xx}sin\left(\frac{x}{\alpha}\right)).\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{4}=-{\alpha}^{2}tsin\left(\frac{x}{\alpha}\right)$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{4}^{t}\hfill & ={\alpha}^{2}sin\left(\frac{x}{\alpha}\right)(2\beta t{\theta}_{tx}{\theta}_{xx}+2\beta t{\theta}_{x}{\theta}_{txx}-2\beta {\theta}_{x}{\theta}_{xx}-t{\theta}_{t}+t{\theta}_{txx}+\theta -{\theta}_{xx}),\hfill \\ {\Omega}_{4}^{x}\hfill & =-{\alpha}^{3}t(-\alpha {\theta}_{xxx}sin\left(\frac{x}{\alpha}\right)+{\theta}_{xx}cos\left(\frac{x}{\alpha}\right)).\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{5}=xt$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{5}^{t}\hfill & =-2\beta xt{\theta}_{tx}{\theta}_{xx}-2\beta xt{\theta}_{txx}{\theta}_{x}+2\beta x{\theta}_{x}{\theta}_{xx}+xt{\theta}_{t}-xt{\theta}_{txx}-x\theta +x{\theta}_{xx},\hfill \\ {\Omega}_{5}^{x}\hfill & =-{\alpha}^{2}xt{\theta}_{xxx}+{\alpha}^{2}t{\theta}_{xx}-xt{\theta}_{x}+t\theta .\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{6}=t$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{6}^{t}\hfill & =-2\beta t{\theta}_{tx}{\theta}_{xx}-2\beta t{\theta}_{txx}{\theta}_{x}+2\beta {\theta}_{x}{\theta}_{xx}+t{\theta}_{t}-t{\theta}_{txx}-\theta +{\theta}_{xx},\hfill \\ {\Omega}_{6}^{x}\hfill & =-{\alpha}^{2}t{\theta}_{xxx}-t{\theta}_{x}.\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{7}=-{\alpha}^{2}cos\left(\frac{x}{\alpha}\right)$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{7}^{t}\hfill & ={\alpha}^{2}cos\left(\frac{x}{\alpha}\right)(2\beta {\theta}_{tx}{\theta}_{xx}+2\beta {\theta}_{x}{\theta}_{txx}-{\theta}_{t}+{\theta}_{txx}),\hfill \\ {\Omega}_{7}^{x}\hfill & ={\alpha}^{3}(\alpha cos\left(\frac{x}{\alpha}\right){\theta}_{xxx}+{\theta}_{xx}sin\left(\frac{x}{\alpha}\right)).\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{8}=-{\alpha}^{2}sin\left(\frac{x}{\alpha}\right)$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{8}^{t}\hfill & ={\alpha}^{2}sin\left(\frac{x}{\alpha}\right)(2\beta {\theta}_{tx}{\theta}_{xx}+2\beta {\theta}_{x}{\theta}_{txx}-{\theta}_{t}+{\theta}_{txx}),\hfill \\ {\Omega}_{8}^{x}\hfill & =-{\alpha}^{3}(-\alpha sin\left(\frac{x}{\alpha}\right){\theta}_{xxx}+{\theta}_{xx}cos\left(\frac{x}{\alpha}\right)).\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{9}=x$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{9}^{t}\hfill & =-2\beta x{\theta}_{tx}{\theta}_{xx}-2\beta x{\theta}_{x}{\theta}_{txx}+x{\theta}_{t}-x{\theta}_{txx}),\hfill \\ {\Omega}_{9}^{x}\hfill & =-{\alpha}^{2}x{\theta}_{xxx}+{\alpha}^{2}{\theta}_{xx}-x{\theta}_{x}+\theta .\hfill \end{array}\right.\end{array}$$
- Using the characteristic ${\Psi}_{10}=1$ in Equation (55) and following the integral Formula (52), we obtain$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Omega}_{10}^{t}\hfill & =-2\beta {\theta}_{tx}{\theta}_{xx}-2\beta {\theta}_{x}{\theta}_{txx}+{\theta}_{t}-{\theta}_{txx}),\hfill \\ {\Omega}_{10}^{x}\hfill & =-{\alpha}^{2}{\theta}_{xxx}-{\theta}_{x}.\hfill \end{array}\right.\end{array}$$

## 5. Wave Nature of the Obtained Solutions

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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$[{\mathcal{E}}_{\mathit{i}},{\mathcal{E}}_{\mathit{j}}]$ | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |
---|---|---|---|---|

${\mathcal{E}}_{1}$ | 0 | 0 | 0 | ${\mathcal{E}}_{2}$ |

${\mathcal{E}}_{2}$ | 0 | 0 | 0 | 0 |

${\mathcal{E}}_{3}$ | 0 | 0 | 0 | 0 |

${\mathcal{E}}_{4}$ | $-{\mathcal{E}}_{2}$ | 0 | 0 | 0 |

$\mathit{Ad}\left({\mathit{e}}^{\mathit{\u03f5}}\right)$ | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |
---|---|---|---|---|

${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}-\u03f5{\mathcal{E}}_{2}$ |

${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |

${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |

${\mathcal{E}}_{4}$ | ${\mathcal{E}}_{1}+\u03f5{\mathcal{E}}_{2}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |

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**MDPI and ACS Style**

Usman, M.; Hussain, A.; Zaman, F.D.; Ibeas, A.; Almalki, Y.
Integrability Properties of the Slepyan–Palmov Model Arising in the Slepyan–Palmov Medium. *Mathematics* **2023**, *11*, 4545.
https://doi.org/10.3390/math11214545

**AMA Style**

Usman M, Hussain A, Zaman FD, Ibeas A, Almalki Y.
Integrability Properties of the Slepyan–Palmov Model Arising in the Slepyan–Palmov Medium. *Mathematics*. 2023; 11(21):4545.
https://doi.org/10.3390/math11214545

**Chicago/Turabian Style**

Usman, Muhammad, Akhtar Hussain, F. D. Zaman, Asier Ibeas, and Yahya Almalki.
2023. "Integrability Properties of the Slepyan–Palmov Model Arising in the Slepyan–Palmov Medium" *Mathematics* 11, no. 21: 4545.
https://doi.org/10.3390/math11214545