TY - BOOK

T1 - Coverings and the fundamental group for partial differential equations

AU - Igonin, S.

PY - 2003

Y1 - 2003

N2 - 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 include constructions like Lax pairs and B\"acklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras $\mathfrak{g}$ we prove that any subalgebra of $\mathfrak{g}$ of finite codimension contains an ideal of $\mathfrak{g}$ of finite codimension.

AB - 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 include constructions like Lax pairs and B\"acklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras $\mathfrak{g}$ we prove that any subalgebra of $\mathfrak{g}$ of finite codimension contains an ideal of $\mathfrak{g}$ of finite codimension.

KW - IR-65863

KW - EWI-3498

KW - MSC-17B65

KW - MSC-37K30

M3 - Report

T3 - Memorandum

BT - Coverings and the fundamental group for partial differential equations

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -