The sine-Gordon equations: Complete and partial integrability

Abstract

The sine–Gordon equation in one space-one time dimension is known to possess the Painlevé property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Bäcklund transform and the Lax pair for this equation. A connection with the sequence of higher-order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painlevé. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painlevé property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painlevé property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space with null scalar curvature.’’ The known integrable solutions satisfy this constraint in a trivial manner. Finally, the coupled KdV, or Hirota–Satsuma, equations possess the Painlevé property. The associated ‘‘modified’’ equations are derived and from these the Lax pair is found.