James D E Grant and I A B Strachan 1999 Nonlinearity 12 1247 doi:10.1088/0951-7715/12/5/302
James D E Grant and I A B Strachan
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In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one of which is in Cauchy-Kovaleskaya form which enables a formal general solution to be given. Various other properties of the field equations and their solutions are studied, such as their symmetry properties and the associated hierarchy of conservation laws.
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws
53A30 Conformal differential geometry
35Q58 Other completely integrable equations (See also 37J35, 37K10)
58Bxx Infinite-dimensional manifolds
37F75 Holomorphic foliations and vector fields (See also 32M25, 32S65, 34Mxx)
Issue 5 (September 1999)
Received 28 August 1998
James D E Grant and I A B Strachan 1999 Nonlinearity 12 1247
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