5,925 research outputs found
Integrable Systems in n-dimensional Riemannian Geometry
In this paper we show that if one writes down the structure equations for the
evolution of a curve embedded in an (n)-dimensional Riemannian manifold with
constant curvature this leads to a symplectic, a Hamiltonian and an hereditary
operator. This gives us a natural connection between finite dimensional
geometry, infinite dimensional geometry and integrable systems. Moreover one
finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries
equation (vmKDV) \vk{t}=
\vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} as integrability condition. We indicate
that other integrable vector evolution equations can be found by using a
different Ansatz on the form of the Lax pair. We obtain these results by using
the {\em natural} or {\em parallel} frame and we show how this can be gauged by
a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If
one chooses the curvature to be zero, as is usual in the context of integrable
systems, then one loses information unless one works in the natural frame
A counterpart of the WKI soliton hierarchy associated with so(3,R)
A counterpart of the Wadati-Konno-Ichikawa (WKI) soliton hierarchy,
associated with so(3,R), is presented through the zero curvature formulation.
Its spectral matrix is defined by the same linear combination of basis vectors
as the WKI one, and its Hamiltonian structures yielding Liouville integrability
are furnished by the trace identity.Comment: 16 page
A counterpart of the WKI soliton hierarchy associated with so(3,R)
A counterpart of the Wadati-Konno-Ichikawa (WKI) soliton hierarchy,
associated with so(3,R), is presented through the zero curvature formulation.
Its spectral matrix is defined by the same linear combination of basis vectors
as the WKI one, and its Hamiltonian structures yielding Liouville integrability
are furnished by the trace identity.Comment: 16 page
A 2 - Component or N=2 Supersymmetric Camassa - Holm Equation
The extended N=2 supersymmetric Camasa - Holm equation is presented. It is
accomplishe by formulation the supersymmeytric version of the Fuchssteiner
method. In this framework we use two supersymmetric recursion operators of the
N=2, Korteweg - de Vries equation and constructed two different
version of the supersymmetric Camassa - Holm equation. The bosonic sector of
N=2, supersymmetric Camassa - Holm equation contains two component
generalization of this equation considered by Chen, Liu and Zhang and as a
special case two component generalized Hunter - Saxton equation considered by
Aratyn, Gomes and Zimerman, As a byproduct of our analysis we defined the N=2
supersymmetric Hunter - Saxton equation. The bihamiltonian structure is
constructed for the supersymmetric N=2, Camassa - Holm equation.Comment: 9 pages, Latex,corrected typo
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