# On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries

## Abstract

**:**

## 1. Introduction

- (1)
- Ibragimov’s conservation law formula is a simple re-writing of a special case of the earlier formula using symmetries and adjoint-symmetries;
- (2)
- Ibragimov’s “nonlinear self-adjointness” condition in its most general form is equivalent to the existence of an adjoint-symmetry for a general DE system and reduces to the existence of a symmetry in the case of a variational DE system;
- (3)
- this formula does not always yield all admitted local conservation laws, and it produces trivial conservation laws whenever the symmetry is a translation and the adjoint-symmetry is translation-invariant;
- (4)
- the computation to find adjoint-symmetries (and, hence, to apply the formula) is just as algorithmic as the computation of local symmetries;
- (5)
- most importantly, if all adjoint-symmetries are known for a given DE system (whether or not it has a variational formulation), then they can be used directly to obtain all local conservation laws, providing a kind of generalization of Noether’s theorem to general DE systems.

## 2. Symmetries, Adjoint-Symmetries and “Nonlinear Self-Adjointness”

#### 2.1. Conservation Laws and Symmetries

#### 2.2. Ibragimov’s Conservation Law Formula

**Proposition**

**1.**

#### 2.3. “Nonlinear Self-Adjointness”

**Theorem**

**1.**

#### 2.4. Adjoint-Symmetries and a Formula for Generating Conservation Laws

**Proposition**

**2.**

**Theorem**

**2.**

## 3. Properties of Conservation Laws Generated by the Adjoint-Symmetry/Symmetry Formula and Ibragimov’s Theorem

**Proposition**

**3.**

**Proposition**

**4.**

#### 3.1. Multiplier Determining equations

**Theorem**

**3.**

**Corollary**

**1.**

#### 3.2. Conservation Laws Produced by a Multiplier/Symmetry Pair

**Theorem**

**4.**

**Corollary**

**2.**

**Theorem**

**5.**

## 4. A Direct Construction Method to Find All Local Conservation Laws

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{Q}(\mathit{t},\mathit{x},\mathit{u})$ | $\mathit{b}\left(\mathit{u}\right)$ | $\mathit{c}\left(\mathit{u}\right)$ | $\mathit{m}\left(\mathit{u}\right)$ | Conditions |
---|---|---|---|---|

${e}^{{m}_{2}t+{m}_{3}x}$ | arb. | arb. | ${m}_{1}u+\int ({m}_{2}b+{m}_{3}c)du$ | ${m}_{1}={m}_{3}^{2}-{m}_{2}^{2}$ |

${e}^{\alpha x+\beta t}$ | ${b}_{0}+{b}_{1}{m}^{\prime}$ | ${c}_{0}+{c}_{1}{m}^{\prime}$ | arb. | $\begin{array}{cc}& {b}_{1}\beta +{c}_{1}\alpha =1\hfill \\ & \beta (\beta -{b}_{0})=\alpha (\alpha +{c}_{0})\hfill \end{array}$ |

${e}^{\gamma x}q(x\mp t)$ | ${b}_{0}+{b}_{1}{m}^{\prime}$ | ${c}_{0}+{c}_{1}{m}^{\prime}$ | arb. | $\begin{array}{cc}& \gamma =\pm {b}_{0}=-{c}_{0},\hfill \\ & {b}_{1}=1/{b}_{0},{c}_{1}=-1/{c}_{0},\hfill \\ & q\left(\xi \right)=\phantom{\rule{4.pt}{0ex}}\mathrm{arb}.\hfill \end{array}$ |

$\mathbf{X}=\mathit{\partial}/\mathit{\partial}\mathit{t}$ | $\mathbf{X}=\mathit{\partial}/\mathit{\partial}\mathit{x}$ | ||
---|---|---|---|

Conditions | $\mathit{Q}$ | ${\mathsf{\Psi}}^{\mathit{t}},{\mathsf{\Psi}}^{\mathit{x}}$ | ${\mathsf{\Psi}}^{\mathit{t}},{\mathsf{\Psi}}^{\mathit{x}}$ |

$\begin{array}{cc}& m={m}_{1}u+{m}_{2}{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du\hfill \\ & \phantom{\rule{2.em}{0ex}}+{m}_{3}{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du\hfill \\ & {m}_{1}={m}_{3}^{2}-{m}_{2}^{2}\hfill \end{array}$ | ${e}^{{m}_{3}x+{m}_{2}t}$ | $\begin{array}{cc}& {m}_{2}Q({u}_{t}-{m}_{2}u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & {m}_{2}Q({m}_{3}u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& {m}_{3}Q({u}_{t}-{m}_{2}u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & {m}_{3}Q({m}_{3}u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ |

$\begin{array}{cc}& b={b}_{0}+{b}_{1}{m}^{\prime}\hfill \\ & c={c}_{0}+{c}_{1}{m}^{\prime}\hfill \\ & {b}_{1}\beta +{c}_{1}\alpha =1\hfill \\ & \beta (\beta -{b}_{0})=\alpha (\alpha +{c}_{0})\hfill \end{array}$ | ${e}^{\alpha x+\beta t}$ | $\begin{array}{cc}& \beta Q({u}_{t}-\beta u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & \beta Q(\alpha u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& \alpha Q({u}_{t}-\beta u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & \alpha Q(\alpha u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ |

$\begin{array}{cc}& b=\pm (\gamma +{\textstyle \frac{1}{\gamma}}{m}^{\prime})\hfill \\ & c=-\gamma +{\textstyle \frac{1}{\gamma}}{m}^{\prime}\hfill \end{array}$ | ${e}^{\gamma x}q(x\mp t)$ | $\begin{array}{cc}& -{e}^{\gamma x}({q}^{\u2033}u\pm {q}^{\prime}({u}_{t}+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)),\hfill \\ & \pm {e}^{\gamma x}((\gamma {q}^{\prime}-{q}^{\u2033})u\hfill \\ & +q({u}_{x}\mp {\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du))\hfill \end{array}$ | $\begin{array}{cc}& {e}^{\gamma x}(\pm ({q}^{\u2033}+\gamma {q}^{\prime})u\hfill \\ & +({q}^{\prime}+\gamma q)({u}_{t}+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)),\hfill \\ & {e}^{\gamma x}(({q}^{\u2033}-{\gamma}^{2}q)u\hfill \\ & -({q}^{\prime}+\gamma q)({u}_{x}\mp {\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du))\hfill \end{array}$ |

Conditions | ${\mathit{Q}}_{\mathit{C}}$ | ${\widehat{\mathit{C}}}^{\mathit{t}}$ | ${\widehat{\mathit{C}}}^{\mathit{x}}$ |
---|---|---|---|

$\begin{array}{cc}& m={m}_{1}u+{m}_{2}{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du\hfill \\ & \phantom{\rule{2.em}{0ex}}+{m}_{3}{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du\hfill \\ & {m}_{1}={m}_{3}^{2}-{m}_{2}^{2}\hfill \end{array}$ | ${e}^{{m}_{3}x+{m}_{2}t}$ | ${e}^{{m}_{3}x+{m}_{2}t}({u}_{t}-{m}_{2}u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)$ | ${e}^{{m}_{3}x+{m}_{2}t}({m}_{3}u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)$ |

$\begin{array}{cc}& b={b}_{0}+{b}_{1}{m}^{\prime}\hfill \\ & c={c}_{0}+{c}_{1}{m}^{\prime}\hfill \\ & {b}_{1}\beta +{c}_{1}\alpha =1\hfill \\ & \beta (\beta -{b}_{0})=\alpha (\alpha +{c}_{0})\hfill \end{array}$ | ${e}^{\alpha x+\beta t}$ | ${e}^{\alpha x+\beta t}({u}_{t}-\beta u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)$ | ${e}^{\alpha x+\beta t}(\alpha u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)$ |

$\begin{array}{cc}& b=\pm (\gamma +{\textstyle \frac{1}{\gamma}}{m}^{\prime})\hfill \\ & c=-\gamma +{\textstyle \frac{1}{\gamma}}{m}^{\prime}\hfill \end{array}$ | ${e}^{\gamma x}q(x\mp t)$ | ${e}^{\gamma x}(q({u}_{t}+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)\pm {q}^{\prime}u)$ | ${e}^{\gamma x}(({q}^{\prime}-\gamma q)u-({u}_{x}\mp {\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du))$ |

P | Q | ${\mathsf{\Psi}}^{\mathit{t}},{\mathsf{\Psi}}^{\mathit{x}}$ | ${\mathit{Q}}_{\mathsf{\Psi}}$ |
---|---|---|---|

$-{u}_{t}$ | ${e}^{{m}_{3}x+{m}_{2}t}$ | $\begin{array}{cc}& {m}_{2}{e}^{{m}_{3}x+{m}_{2}t}({u}_{t}-{m}_{2}u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & {m}_{2}{e}^{{m}_{3}x+{m}_{2}t}({m}_{3}u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& {m}_{2}{e}^{{m}_{3}x+{m}_{2}t}\hfill \\ & ={D}_{t}\left({e}^{{m}_{3}x+{m}_{2}t}\right)\hfill \end{array}$ |

$-{u}_{x}$ | ${e}^{{m}_{3}x+{m}_{2}t}$ | $\begin{array}{cc}& {m}_{3}{e}^{{m}_{3}x+{m}_{2}t}({u}_{t}-{m}_{2}u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & {m}_{3}{e}^{{m}_{3}x+{m}_{2}t}({m}_{3}u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& {m}_{3}{e}^{{m}_{3}x+{m}_{2}t}\hfill \\ & ={D}_{x}\left({e}^{{m}_{3}x+{m}_{2}t}\right)\hfill \end{array}$ |

$-{u}_{t}$ | ${e}^{\alpha x+\beta t}$ | $\begin{array}{cc}& \beta {e}^{\alpha x+\beta t}({u}_{t}-\beta u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & \beta {e}^{\alpha x+\beta t}(\alpha u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& \beta {e}^{\alpha x+\beta t}\hfill \\ & ={D}_{t}\left({e}^{\alpha x+\beta t}\right)\hfill \end{array}$ |

$-{u}_{x}$ | ${e}^{\alpha x+\beta t}$ | $\begin{array}{cc}& \alpha {e}^{\alpha x+\beta t}({u}_{t}-\beta u+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du),\hfill \\ & \alpha {e}^{\alpha x+\beta t}(\alpha u-{u}_{x}+{\textstyle \int}c\phantom{\rule{0.166667em}{0ex}}du)\hfill \end{array}$ | $\begin{array}{cc}& \alpha {e}^{\alpha x+\beta t}\hfill \\ & ={D}_{x}\left({e}^{\alpha x+\beta t}\right)\hfill \end{array}$ |

$-{u}_{t}$ | ${e}^{\gamma x}q$ | $\begin{array}{cc}& -{e}^{\gamma x}({q}^{\u2033}u\pm {q}^{\prime}({u}_{t}+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)),\hfill \\ & \pm {e}^{\gamma x}((\gamma {q}^{\prime}-{q}^{\u2033})u+q({u}_{x}\mp {\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du))\hfill \end{array}$ | $\begin{array}{cc}& \mp {e}^{\gamma x}q\hfill \\ & ={D}_{t}\left({e}^{\gamma x}q\right)\hfill \end{array}$ |

$-{u}_{x}$ | ${e}^{\gamma x}q$ | $\begin{array}{cc}& {e}^{\gamma x}(\pm ({q}^{\u2033}+\gamma {q}^{\prime})u+({q}^{\prime}+\gamma q)({u}_{t}+{\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du)),\hfill \\ & {e}^{\gamma x}(({q}^{\u2033}-{\gamma}^{2}q)u-({q}^{\prime}+\gamma q)({u}_{x}\mp {\textstyle \int}b\phantom{\rule{0.166667em}{0ex}}du))\hfill \end{array}$ | $\begin{array}{cc}& {e}^{\gamma x}({q}^{\prime}+\gamma q)\hfill \\ & ={D}_{x}\left({e}^{\gamma x}q\right)\hfill \end{array}$ |

Conditions | ${\mathit{Q}}_{\mathit{C}}$ |
---|---|

$\begin{array}{cc}& \frac{2{m}_{1}{m}_{2}}{m-{m}_{1}}=\frac{4{m}_{1}}{b\pm c}={\textstyle \int}(b\mp c)\phantom{\rule{0.166667em}{0ex}}du\hfill \end{array}$ | $\frac{2{m}_{1}+(b\pm c)({u}_{t}\pm {u}_{x})}{2m+(b\pm c)({u}_{t}\pm {u}_{x})}$ |

$\begin{array}{cc}& m=({m}_{1}+{\textstyle \frac{1}{4}}{\textstyle \int}(b-c)\phantom{\rule{0.166667em}{0ex}}du)(b+c),\hfill \\ & (1-\gamma )b=(1+\gamma )c\hfill \end{array}$ | $\frac{((1-\gamma ){u}_{t}+(1+\gamma ){u}_{x})({b}^{2}-{c}^{2})}{((b+c)({u}_{t}+{u}_{x})+2m)\left((b-c)({u}_{t}-{u}_{x})2m\right)}$ |

Conditions | ${\widehat{\mathit{C}}}^{\mathit{t}}$, ${\widehat{\mathit{C}}}^{\mathit{x}}$ |
---|---|

$\begin{array}{cc}& \frac{2{m}_{1}{m}_{2}}{m-{m}_{1}}=\frac{4{m}_{1}}{b\pm c}={\textstyle \int}(b\mp c)\phantom{\rule{0.166667em}{0ex}}du\hfill \end{array}$ | $\begin{array}{cc}& \gamma ln\left(\frac{b\pm c}{2{m}_{1}+(b\pm c)(\gamma +{u}_{t}\pm {u}_{x})}\right)+{u}_{t}+{\textstyle \frac{1}{2}}{\textstyle \int}(b\pm c)\phantom{\rule{0.166667em}{0ex}}du,\hfill \\ & \mp \gamma ln\left(\frac{b\pm c}{2{m}_{1}+(b\pm c)(\gamma +{u}_{t}\pm {u}_{x})}\right)-{u}_{x}+{\textstyle \frac{1}{2}}{\textstyle \int}(c\pm b)\phantom{\rule{0.166667em}{0ex}}du+\gamma x\hfill \end{array}$ |

$\begin{array}{cc}& m=\left({m}_{1}+{\textstyle \frac{1}{4}}{\textstyle \int}(b-c)\phantom{\rule{0.166667em}{0ex}}du\right)(b+c),\hfill \\ & (1-\gamma )b=(1+\gamma )c\hfill \end{array}$ | $\begin{array}{cc}& ln\left(\frac{{\left(\gamma \left({\textstyle \int}(b+c)\phantom{\rule{0.166667em}{0ex}}du+2({u}_{t}-{u}_{x})\right)+{m}_{1}\right)}^{\frac{1}{\gamma}}}{\gamma {\textstyle \int}(b+c)\phantom{\rule{0.166667em}{0ex}}du+2({u}_{t}+{u}_{x})+{m}_{1}}\right),\hfill \\ & ln({\left(\gamma \left({\textstyle \int}(b+c)\phantom{\rule{0.166667em}{0ex}}du+2({u}_{t}-{u}_{x})\right)+{m}_{1}\right)}^{\frac{1}{\gamma}}\hfill \\ & \phantom{\rule{2.em}{0ex}}\times \left(\gamma {\textstyle \int}(b+c)\phantom{\rule{0.166667em}{0ex}}du+2({u}_{t}+{u}_{x})+{m}_{1}\right))\hfill \end{array}$ |

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**MDPI and ACS Style**

Anco, S.C.
On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries. *Symmetry* **2017**, *9*, 33.
https://doi.org/10.3390/sym9030033

**AMA Style**

Anco SC.
On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries. *Symmetry*. 2017; 9(3):33.
https://doi.org/10.3390/sym9030033

**Chicago/Turabian Style**

Anco, Stephen C.
2017. "On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries" *Symmetry* 9, no. 3: 33.
https://doi.org/10.3390/sym9030033