FoCM 2014 conference

Workshop B7 - Symbolic Analysis

December 15, 18:00 ~ 18:25 - Room A11

## Nonlocal symmetries and formal integrability

### Enrique G. Reyes

### Universidad de Santiago de Chile, Chile - ereyes@fermat.usach.cl ; e_g_reyes@yahoo.ca

In this talk I introduce a simplified version of the classical Krasil'shchik-Vinogradov geometric theory of nonlocal symmetries and present several applications. For example, the theory can be used to find highly non-trivial explicit solutions and Darboux-like transforms to nonlinear equations such as the Kaup-Kupershmidt equation. I also recall the theory of formal integrability and argue that nonlocal symmetries can be used to uncover formally integrable equations. Finally, I present some classifications of nonlocal symmetries of integrable equations which have been recently found, and propose a generalization of the Krasil'shchik-Vinogradov theory.

\smallskip

This talk is partially based on the following papers:

\smallskip

1. E.G. Reyes, Geometric integrability of the Camassa-Holm equation. Letters in Mathematical Physics 59 (2002), 117--131.

2. E.G. Reyes, Nonlocal symmetries and the Kaup-Kupershmidt equation. Journal of Mathematical Physics 46 (2005), 073507 (19 pages).

3. P. Gorka and E.G. Reyes, The modified Camassa-Holm equation. International Mathematics Research Notices (2011) Vol. 2011, 2617--2649.

4. R. Hernandez-Heredero and E.G. Reyes, Geometric integrability of the Camassa-Holm equation II. International Mathematics Research Notices (2012) Vol. 2012, 3089--3125.

5. E.G. Reyes, Jet bundles, symmetries, Darboux transforms. Contemporary Mathematics 563 (2012), 137--164.

6. P. Gorka and E.G. Reyes, The modified Hunter-Saxton equation. Journal of Geometry and Physics 62 (2012), 1793--1809.

7. P.M. Bies, P. Gorka and E.G. Reyes, The dual modified KdV--Fokas--Qiao equation: geometry and local analysis. Journal of Mathematical Physics 53 (2012), 073710.

8. R. Hernandez-Heredero and E.G. Reyes, Nonlocal symmetries, compacton equations, and integrability. International Journal of Geometric Methods in Modern Physics 10 (2013), 1350046 [24 pages].

9. I.S. Krasil'shchik and A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws and Backlund transformations. Acta Appl. Math. 15 (1989), 161--209.