# Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method

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## Abstract

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## 1. Introduction

## 2. Fundamentals of the Double Reduction Theorem

- The total derivative operator with respect to ${x}^{i}$ is$${D}_{i}=\frac{\partial}{\partial {x}^{i}}+{u}_{i}\frac{\partial}{\partial u}+{u}_{ij}\frac{\partial}{\partial {u}_{j}}+\cdots ,\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,n,\phantom{\rule{-3.0pt}{0ex}}$$
- The determining equations for multipliers are obtained by taking the variational derivative$$\frac{\delta}{\delta u}\left(\mathsf{\Lambda}E\right)=0,$$$$\frac{\delta}{\delta u}=\frac{\partial}{\partial u}-{D}_{i}\frac{\partial}{\partial {u}_{i}}+{D}_{ij}\frac{\partial}{\partial {u}_{ij}}-{D}_{ijk}\frac{\partial}{\partial {u}_{ijk}}+\cdots .$$
- A Lie symmetry of (2) with infinitesimal generator $X={\xi}_{i}\partial {x}_{i}+\eta \partial u$ is said to be associated with a conserved law (4) if the symmetry and the conservation law satisfy the relations [16]$$\left[{T}^{i},X\right]=X\left({T}^{i}\right)+{T}^{i}{D}_{j}{\xi}^{j}-{T}^{j}{D}_{j}{\xi}^{i},\phantom{\rule{1.em}{0ex}}i=1,\dots ,n.$$

- I.
- Find similarity variables ${\tilde{x}}_{i},i=1,2,\dots ,n$ and w,$$\begin{array}{ccc}\hfill {\tilde{x}}_{i}& =& {\tilde{x}}_{i}\left({x}^{1},{x}^{2},\dots ,{x}^{n}\right),\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,n\hfill \\ \hfill w\left({\tilde{x}}_{1},\dots ,{\tilde{x}}_{n-1}\right)& =& \omega \left({x}^{1},{x}^{2},\dots ,{x}^{n}\right)u,\hfill \end{array}$$
- II.
- Find inverse canonical coordinates$$\begin{array}{ccc}\hfill {x}^{i}& =& {x}^{i}\left({\tilde{x}}_{1},{\tilde{x}}_{2},\dots ,{\tilde{x}}_{n}\right),\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,n\hfill \\ \hfill u\left({x}^{1},{x}^{2},\dots ,{x}^{n}\right)& =& \psi \left({\tilde{x}}_{1},{\tilde{x}}_{2},\dots ,{\tilde{x}}_{n}\right)w.\hfill \end{array}$$
- III.
- Write partial derivatives of u in terms of the similarity variables.
- IV.
- Construct matrices A and ${A}^{-1}$ as follows:$$A=\left(\begin{array}{cccc}{\tilde{D}}_{1}{x}_{1}& {\tilde{D}}_{1}{x}_{2}& \dots & {\tilde{D}}_{1}{x}_{n}\\ {\tilde{D}}_{2}{x}_{1}& {\tilde{D}}_{2}{x}_{2}& \dots & {\tilde{D}}_{2}{x}_{n}\\ \vdots & \vdots & \vdots & \vdots \\ {\tilde{D}}_{n}{x}_{1}& {\tilde{D}}_{n}{x}_{2}& \dots & {\tilde{D}}_{n}{x}_{n}\end{array}\right),\phantom{\rule{1.em}{0ex}}{A}^{-1}=\left(\begin{array}{cccc}{D}_{1}{\tilde{x}}_{1}& {D}_{1}{\tilde{x}}_{2}& \dots & {D}_{1}{\tilde{x}}_{n}\\ {D}_{2}{\tilde{x}}_{1}& {D}_{2}{\tilde{x}}_{2}& \dots & {D}_{2}{\tilde{x}}_{n}\\ \vdots & \vdots & \vdots & \vdots \\ {D}_{n}{\tilde{x}}_{1}& {D}_{n}{\tilde{x}}_{2}& \dots & {D}_{n}{\tilde{x}}_{n}\end{array}\right).$$
- V.
- Write components ${T}^{i}$ of the conserved vector in terms of the similarity variables as follows:$$\left(\begin{array}{c}{\tilde{T}}^{1}\\ {\tilde{T}}^{2}\\ \vdots \\ {\tilde{T}}^{n}\end{array}\right)=J{\left({A}^{-1}\right)}^{T}\left(\begin{array}{c}{T}^{1}\\ {T}^{2}\\ \vdots \\ {T}^{n}\end{array}\right),$$
- VI.
- The reduced conservation law becomes$${D}_{1}{\tilde{T}}^{1}+{D}_{2}{\tilde{T}}^{2}+\cdots +{D}_{n-1}{\tilde{T}}^{n-1}=0.$$

## 3. Symmetries and Conservation Laws of the Hunter–Saxton Equation

## 4. Double Reduction of the Hunter–Saxton Equation

#### 4.1. Double Reduction of (1) by $\u2329{\kappa}_{1}({X}_{1}+2{X}_{3})+{\kappa}_{2}{X}_{2}\u232a$

#### 4.2. Double Reduction of (1) by $\u2329{\kappa}_{1}({X}_{1}+{X}_{3})+{\kappa}_{2}{X}_{2}\u232a$

#### 4.3. Double Reduction of (1) by $\u2329{\kappa}_{1}\left({X}_{1}-\frac{{X}_{3}}{2}\right)+{\kappa}_{2}{X}_{2}\u232a$

#### 4.4. Double Reduction of (1) by $\u2329{X}_{3}\u232a$

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kakuli, M.C.; Sinkala, W.; Masemola, P.
Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method. *Math. Comput. Appl.* **2023**, *28*, 92.
https://doi.org/10.3390/mca28050092

**AMA Style**

Kakuli MC, Sinkala W, Masemola P.
Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method. *Mathematical and Computational Applications*. 2023; 28(5):92.
https://doi.org/10.3390/mca28050092

**Chicago/Turabian Style**

Kakuli, Molahlehi Charles, Winter Sinkala, and Phetogo Masemola.
2023. "Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method" *Mathematical and Computational Applications* 28, no. 5: 92.
https://doi.org/10.3390/mca28050092