APPLICATIONS OF THE SINGULAR MANIFOLD METHOD

John Weiss

Applied Mathematics Group,
49 Grand View Road, Arlington, MA 02476, USA
E-mail: johnweiss@rcn.com
Web: www.appmath.com

For systems without the Painlevé Property, it is possible to constrain the arbitrary functions in the expansion so as to restore the single-valued behavior. Depending on the number of constraints, the resulting expansion will represent a solution of reduced dimensions. The systems of constraints are expressed as a system of partial differential equations for the previously arbitrary functions in the expansion. We conjecture that the systems of constraint equations are completely integrable. Therefore, the analysis of differential equations by the Singular Manifold Method could lead to the discovery of new integrable systems. This is especially interesting for systems depending on many variables. We present some new results for the equations defined by the singular manifold analysis of the Sine-Gordon equation in several variables.