10,088 research outputs found
Lenard scheme for two dimensional periodic Volterra chain
We prove that for compatible weakly nonlocal Hamiltonian and symplectic
operators, hierarchies of infinitely many commuting local symmetries and
conservation laws can be generated under some easily verified conditions no
matter whether the generating Nijenhuis operators are weakly nonlocal or not.
We construct a recursion operator of the two dimensional periodic Volterra
chain from its Lax representation and prove that it is a Nijenhuis operator.
Furthermore we show this system is a (generalised) bi-Hamiltonian system.
Rather surprisingly, the product of its weakly nonlocal Hamiltonian and
symplectic operators gives rise to the square of the recursion operator.Comment: Submit to Journal of Mathematical Physic
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
ANALYSIS ON THE COMMONNESS BETWEEN COLLEGE STUDENTS’ MENTAL HEALTH PROBLEMS AND IDEOLOGICAL AND POLITICAL EDUCATION
ANALYSIS ON THE COMMONNESS BETWEEN COLLEGE STUDENTS’ MENTAL HEALTH PROBLEMS AND IDEOLOGICAL AND POLITICAL EDUCATION
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