2,710 research outputs found
Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves
We consider integrable discretizations of some soliton equations associated
with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam
equation, the complex Dym equation, and the short pulse equation. They are
related to the modified KdV or the sine-Gordon equations by the hodograph
transformations. Based on the observation that the hodograph transformations
are regarded as the Euler-Lagrange transformations of the curve motions, we
construct the discrete analogues of the hodograph transformations, which yield
integrable discretizations of those soliton equations.Comment: 19 page
Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy
The equation of a motion of curves in the projective plane is deduced. Local
flows are defined in terms of polynomial differential functions. A family of
local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The
integration of the congruence curves is discussed. Local motions defined by the
traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation
are described
Hamiltonian flows on null curves
The local motion of a null curve in Minkowski 3-space induces an evolution
equation for its Lorentz invariant curvature. Special motions are constructed
whose induced evolution equations are the members of the KdV hierarchy. The
null curves which move under the KdV flow without changing shape are proven to
be the trajectories of a certain particle model on null curves described by a
Lagrangian linear in the curvature. In addition, it is shown that the curvature
of a null curve which evolves by similarities can be computed in terms of the
solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio
A non-linear Oscillator with quasi-Harmonic behaviour: two- and -dimensional Oscillators
A nonlinear two-dimensional system is studied by making use of both the
Lagrangian and the Hamiltonian formalisms. The present model is obtained as a
two-dimensional version of a one-dimensional oscillator previously studied at
the classical and also at the quantum level. First, it is proved that it is a
super-integrable system, and then the nonlinear equations are solved and the
solutions are explicitly obtained. All the bounded motions are quasiperiodic
oscillations and the unbounded (scattering) motions are represented by
hyperbolic functions. In the second part the system is generalized to the case
of degrees of freedom. Finally, the relation of this nonlinear system with
the harmonic oscillator on spaces of constant curvature, two-dimensional sphere
and hyperbolic plane , is discussed.Comment: 30 pages, 4 figures, submitted to Nonlinearit
Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
The bi-Hamiltonian structure of the two known vector generalizations of the
mKdV hierarchy of soliton equations is derived in a geometrical fashion from
flows of non-stretching curves in Riemannian symmetric spaces G/SO(N). These
spaces are exhausted by the Lie groups G=SO(N+1),SU(N). The derivation of the
bi-Hamiltonian structure uses a parallel frame and connection along the curves,
tied to a zero curvature Maurer-Cartan form on G, and this yields the vector
mKdV recursion operators in a geometric O(N-1)-invariant form. The kernel of
these recursion operators is shown to yield two hyperbolic vector
generalizations of the sine-Gordon equation. The corresponding geometric curve
flows in the hierarchies are described in an explicit form, given by wave map
equations and mKdV analogs of Schrodinger map equations.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA/ Minor changes made (typos
corrected and more discussion added about parallel frames and vector SG
equations
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