A non-Archimedean approach to prolongation theory

  title={A non-Archimedean approach to prolongation theory},
  author={H. N. van Eck},
  journal={Letters in Mathematical Physics},
  • H. Eck
  • Published 1986
  • Mathematics
  • Letters in Mathematical Physics
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. 
Lie algebras responsible for zero-curvature representations of scalar evolution equations
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
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, MKdVExpand
On a valuation field invented by A. Robinson and certain structures connected with it
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 thisExpand
Infinite-dimensional prolongation Lie algebras and multicomponent Landau–Lifshitz systems associated with higher genus curves
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 generalExpand
Coverings and the Fundamental Group 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 differentialExpand
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 differentialExpand
Coverings and the fundamental group for partial differential equations
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, whichExpand
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 representationsExpand
Coverings and fundamental algebras for partial differential equations
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 asExpand
On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
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 ofExpand


The explicit structure of the nonlinear Schrödinger prolongation algebra
The structure of the nonlinear Schrodinger prolongation algebra, introduced by Estabrook and Wahlquist, is explicitly determined. It is proved that this Lie algebra is isomorphic with the directExpand
Asymptotic Behaviour of the Resolvent of Sturm-Liouville Equations and the Algebra of the Korteweg-De Vries Equations
This paper is concerned with a group of problems associated with recent results on non-linear equations of Korteweg-de Vries type. The second chapter is basically of a survey character andExpand
Prolongation structures of nonlinear evolution equations
The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubicallyExpand
The explicit form of the Lie algebra of Wahlquist and Estabrook. A presentation problem
The structure of the KdV-Lie algebra of Wahlquist and Estabrook is made explicit. This is done with help of a table of Lie-products and an inherent grading of the algebra.
Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”
For a systemY of partial differential equations, the notion of a coveringŶ∞→Y∞ is introduced whereY∞ is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations ofŶ∞Expand
Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and relatedExpand
The Estabrook-Wahlquist method with examples of application
Abstract The Estabrook-Wahlquist method can be used as a semidirect method for starting with a given nonlinear partial differential equation, and if it would happen to be exactly solvable, allowingExpand
Local symmetries and conservation laws
Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessaryExpand
Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
An explicit nonlinear transformation relating solutions of the Korteweg‐de Vries equation and a similar nonlinear equation is presented. This transformation is generalized to solutions of aExpand
Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrödinger equations
Group theoretic properties of nonlinear time evolution equations have been studied from the standpoint of a generalized Lie transformation. It has been found that with each constant of motion of theExpand