Computer-Aided Closure of the Lie Algebra Associated with a Nonlinear PDE,
Abstract
Since the solution of the KdV equation by Gardner, Greene, Kruskal and Miura in 1967, the inverse scattering method which they discovered has been extended to solve a large class of nonlinear PDE's. However, a method for seeking an inverse scattering solution to a given equation has emerged from the work of Wahlquist and Estabrook. The Wahlquist-Estabrook (WE) method systematically associates a set of Lie brackets with a given nonlinear PDE. If this Lie algebra can be closed, consistent with the Jacobi identity, then an associated linear representation gives the inverse scattering transform, when one exists. Even when an inverse scattering transform does not exist, if the Lie algebra can be closed the associated linear representation can provide useful information about the solutions of the nonlinear PDE. This task involves repeated application of the Jacobi identities, which in turn requires the evaluation of a large number of Lie brackets which must be looked up in a table. The simple but repetitive nature of this task suggests that we attempt to mechanize it using a computer. However, we must also require that our program automatically perform algebraic simplifications at each step: properly distributing the Lie bracket over a sum of terms, combining like terms, multiplying numerical coefficients, etc. We have therefore used the symbolic manipulation language MACSYMA to implement our algorithms. The language has built-in capabilities for performing the standard simple algebraic manipulations from which we build up our algorithms.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1980
- Accession Number
- ADA093640
Entities
People
- Allan Reiman
Organizations
- University of Maryland