The sine-Gordon equations: Complete and partial integrability

Journal Article · Sun Jul 01 04:00:00 EDT 1984 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.526415· OSTI ID:6741385

Weiss, J

The sine--Gordon equation in one space-one time dimension is known to possess the Painleve property and to be completely integrable. It is shown how the method of ''singular manifold'' analysis obtains the Baecklund 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 Painleve. Also, certain ''rational'' solutions are constructed iteratively. The double sine--Gordon equation is shown not to possess the Painleve 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 Painleve 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 Painleve property. The associated ''modified'' equations are derived and from these the Lax pair is found.

• Research Organization: La Jolla Institute, La Jolla, California 92037; and Institute for Pure and Applied Physical Science, University of California, San Diego, La Jolla, California 92093
• DOE Contract Number: AC03-81ER10923
• OSTI ID: 6741385
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 25:7; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

645400* -- High Energy Physics-- Field Theory
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ANALYTICAL SOLUTION
BAECKLUND TRANSFORMATION
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EQUATIONS
ITERATIVE METHODS
KORTEWEG-DE VRIES EQUATION
LAX THEOREM
MATHEMATICAL MANIFOLDS
MODIFICATIONS
PARTIAL DIFFERENTIAL EQUATIONS
SINE-GORDON EQUATION
SPACE-TIME
TRANSFORMATIONS