# A non-Archimedean approach to prolongation theory

@article{Eck1986ANA, title={A non-Archimedean approach to prolongation theory}, author={H. N. van Eck}, journal={Letters in Mathematical Physics}, year={1986}, volume={12}, pages={231-239} }

Some evolution equations possess infinite-dimensional prolongation Lie algebras which can be made finite-dimensional by using a bigger (non-Archimedean) field. The advantage of this is that convergence problems hardly exist in such a field. Besides that, the accompanying Lie groups can be easily constructed.

#### 14 Citations

Lie algebras responsible for zero-curvature representations of scalar evolution equations

- Mathematics, Physics
- Journal of Geometry and Physics
- 2019

Zero-curvature representations (ZCRs) are well known to be one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.

Note on Backlund transformations

- Mathematics
- 1987

The method of obtaining Backlund transformations proposed by Chern and Tenenblat (1986) fits completely the approach of obtaining Backlund transformations by prolongation techniques. For KdV, MKdV… Expand

On a valuation field invented by A. Robinson and certain structures connected with it

- Mathematics
- 1991

We clarify the structure of the non-archimedean valuation fieldρR which was introduced by A. Robinson, and of theρ-non-archimedean hulls of Banach algebras and Lie groups. (For Banach spaces this… Expand

Infinite-dimensional prolongation Lie algebras and multicomponent Landau–Lifshitz systems associated with higher genus curves

- Mathematics, Physics
- 2013

Abstract The Wahlquist–Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund transformations of certain type. We present some general… Expand

Coverings and the Fundamental Group for Partial Differential Equations

- 2004

Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special mor-phisms called differential… Expand

Coverings and Fundamental Algebras for Partial Differential Equations

Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special mor-phisms called differential… Expand

Coverings and the fundamental group for partial differential equations

- Physics, Mathematics
- 2003

Following I. S. Krasilshchik and A. M. Vinogradov, we regard systems of PDEs as manifolds with involutive distributions and consider their special morphisms called differential coverings, which… Expand

HIGHER JET PROLONGATION LIE ALGEBRAS AND B ¨ ACKLUND TRANSFORMATIONS FOR (1 + 1)-DIMENSIONAL PDES

- Mathematics, Physics
- 2013

For any (1+1)-dimensional (multicomponent) evolution PDE, we define a sequence of Lie algebras $F^p$, $p=0,1,2,3,...$, which are responsible for all Lax pairs and zero-curvature representations… Expand

Coverings and fundamental algebras for partial differential equations

- Mathematics, Physics
- 2006

Abstract Following Krasilshchik and Vinogradov [I.S. Krasilshchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations, Acta Appl. Math. 15 (1989) 161–209], we regard PDEs as… Expand

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

- Physics, Mathematics
- 2020

Abstract Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13] , for any (1+1)-dimensional scalar evolution equation E , we defined a family of… Expand

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