#
Nonlocal Conservation Laws of PDEs Possessing Differential Coverings^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Remark**

**1.**

**Coordinates.**Consider a trivialization of $\pi $ with local coordinates ${x}^{1},\cdots ,{x}^{n}$ in $\mathcal{U}\subset M$ and ${u}^{1},\cdots ,{u}^{m}$ in the fibers of ${\left.\pi \right|}_{\mathcal{U}}$. Then in ${\pi}_{\infty}^{-1}\left(\mathcal{U}\right)\subset {J}^{\infty}\left(\pi \right)$ the adapted coordinates ${u}_{\sigma}^{i}$ arise and the Cartan connection is determined by the total derivatives

**Coordinates.**Choose a trivialization of the covering $\tau $ and let ${w}^{1},\cdots ,{w}^{l},\cdots $ be coordinates in fibers (the are called nonlocal variables). Then the covering structure is given by the extended total derivatives

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 3. The Main Result

**Example**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proposition**

**3.**

**Coordinates.**Let $rank\left(\tau \right)=l>1$ and

**Theorem**

**1.**

**Proof.**

- $l>{l}_{0}$, because $\tau $ is a nontrivial covering;
- the integral leaves project to $\mathcal{E}$ surjectively, because $\mathcal{E}$ is a differentially connected equation.

## 4. Examples

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bianchi, L. Sulla trasformazione di Bäcklund per le superfici pseudosferiche. Rend. Mat. Acc. Lincei
**1892**, 1, 3–12. [Google Scholar] - Chatterjee, N.; Fjordholm, U.S. A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws. IMA J. Numer. Anal.
**2020**, 40, 405–421. [Google Scholar] [CrossRef] - Naz, R. Potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere. Z. Naturforschung A
**2017**, 72, 351–357. [Google Scholar] [CrossRef] - Aggarwal, A.; Goatin, P. Crowd dynamics through non-local conservation laws. Bull. Braz. Math. Soc. New Ser.
**2016**, 47, 37–50. [Google Scholar] [CrossRef] [Green Version] - Keimer, A.; Pflug, L. Nonlocal conservation laws with time delay. Nonlinear Differ. Equ. Appl.
**2019**, 26, 54. [Google Scholar] [CrossRef] - Sil, S.; Sekhar, T.R.; Zeidan, D. Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation. Chaos Solitons Fractals
**2020**, 139, 110010. [Google Scholar] [CrossRef] - Anco, S.C.; Bluman, G. Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations. J. Math. Phys.
**1997**, 38, 350. [Google Scholar] [CrossRef] [Green Version] - Anco, S.C.; Webb, G.M. Conservation laws in magnetohydrodynamics and fluid dynamics: Lagrangian approach. AIP Conf. Proc.
**2019**, 2153, 020024. [Google Scholar] - Betancourt, F.; Bürger, R.; Karlsen, K.; Tory, E.M. On nonlocal conservation laws modelling sedimentation. Nonlinearity
**2011**, 24, 855–885. [Google Scholar] [CrossRef] - Christoforou, C. Nonlocal conservation laws with memory. In Hyperbolic Problems: Theory, Numerics, Applications; Benzoni-Gavage, S., Serre, D., Eds.; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ibragimov, N.; Karimova, E.N.; Galiakberova, L.R. Chaplygin gas motions associated with nonlocal conservation laws. J. Coupled Syst. Multiscale Dyn.
**2017**, 5, 63–68. [Google Scholar] [CrossRef] - Krasil’shchik, I.S.; Vinogradov, A.M. Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math.
**1989**, 15, 161–209. [Google Scholar] - Bocharov, A.V.; Chetverikov, V.N.; Duzhin, S.V.; Khor’kova, N.G.; Krasil’shchik, I.S.; Samokhin, A.V.; Torkhov, Y.N.; Verbovetsky, A.M.; Vinogradov, A.M. Symmetries of Differential Equations in Mathematical Physics and Natural Sciences; English translation: Amer. Math. Soc., 1999; Vinogradov, A.M., Krasil’shchik, I.S., Eds.; Factorial Publ. House: Moscow, Russia, 1997. (In Russian) [Google Scholar]
- Vinogradov, A.M. Cohomological Analysis of Partial Differential Equations and Secondary Calculus; Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 2001; Volume 204. [Google Scholar]
- Krasil’shchik, I.S.; Verbovetskiy, A.M.; Vitolo, R. The Symbolic Computation of Integrability Structures for Partial Differential Equations; Texts & Monographs in Symbolic Computation; Springer: Berlin, Germnay, 2017. [Google Scholar]
- Gardner, C.S.; Green, J.M.; Kruskal, M.D.; Miura, R.M. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett.
**1967**, 19, 1095–1097. [Google Scholar] [CrossRef] - Lax, P.D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.
**1968**, 21, 467. [Google Scholar] [CrossRef] - Wahlquist, H.B.; Estabrook, F.B. Bäcklund transformation for solutions to the Korteweg-de Vries equation. Phys. Rev. Lett.
**1973**, 31, 1386–1390. [Google Scholar] [CrossRef] - Sym, A. Soliton surfaces and their applications (soliton geometry from spectral problems). In Geometric Aspects of the Einstein Equations and Integrable Systems, Proceedings of the Conference Scheveningen, The Netherlands, 26–31 August 1984; Lecture Notes in Physics; Martini, R., Ed.; Springer: Berlin, Germany, 1985; Volume 239, pp. 154–231. [Google Scholar]
- Krasil’shchik, I.S.; Marvan, M. Coverings and integrability of the Gauss-Mainardi-Codazzi equations. Acta Appl. Math.
**1999**, 56, 217–230. [Google Scholar] - Pavlov, M.V. Integrable hydrodynamic chains. J. Math. Phys.
**2003**, 44, 4134. [Google Scholar] [CrossRef] [Green Version] - Baran, H.; Krasil’shchik, I.S.; Morozov, O.I.; Vojčák, P. Nonlocal symmetries of integrable linearly degenerate equations: A comparative study. Theoret. Math. Phys.
**2018**, 196, 169–192. [Google Scholar] [CrossRef]

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Krasil’shchik, I.
Nonlocal Conservation Laws of PDEs Possessing Differential Coverings. *Symmetry* **2020**, *12*, 1760.
https://doi.org/10.3390/sym12111760

**AMA Style**

Krasil’shchik I.
Nonlocal Conservation Laws of PDEs Possessing Differential Coverings. *Symmetry*. 2020; 12(11):1760.
https://doi.org/10.3390/sym12111760

**Chicago/Turabian Style**

Krasil’shchik, Iosif.
2020. "Nonlocal Conservation Laws of PDEs Possessing Differential Coverings" *Symmetry* 12, no. 11: 1760.
https://doi.org/10.3390/sym12111760