5,486 research outputs found
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 New Soliton Hierarchy Associated with so
Based on the three-dimensional real special orthogonal Lie algebra so(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy
A Counterpart of the Wadati-Konno-Ichikawa 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
Conformal invariant functionals of immersions of tori into R^3
We show, that higher analogs of the Willmore functional, defined on the space
of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is
the 3-dimensional Euclidean space are invariant under conformal transformations
of R^3. This hypothesis was formulated recently by I.A.Taimanov
(dg-ga/9610013).
Higher analogs of the Willmore functional are defined in terms of the
Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with
the zero-energy scattering problem for the two-dimensional Dirac operator.Comment: 34 pages, LaTeX, amssym.def macros use
A geometric interpretation of the spectral parameter for surfaces of constant mean curvature
Considering the kinematics of the moving frame associated with a constant
mean curvature surface immersed in S^3 we derive a linear problem with the
spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral
parameter is related to the radius R of the sphere S^3. The application of the
Sym formula to this linear problem yields constant mean curvature surfaces in
E^3. Independently, we show that the Sym formula itself can be derived by an
appropriate limiting process R -> infinity.Comment: 12 page
-Strands
A -strand is a map for a Lie
group that follows from Hamilton's principle for a certain class of
-invariant Lagrangians. The SO(3)-strand is the -strand version of the
rigid body equation and it may be regarded physically as a continuous spin
chain. Here, -strand dynamics for ellipsoidal rotations is derived as
an Euler-Poincar\'e system for a certain class of variations and recast as a
Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as
for a perfect complex fluid. For a special Hamiltonian, the -strand is
mapped into a completely integrable generalization of the classical chiral
model for the SO(3)-strand. Analogous results are obtained for the
-strand. The -strand is the -strand version of the
Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical
sorting. Numerical solutions show nonlinear interactions of coherent wave-like
solutions in both cases. -strand equations on the
diffeomorphism group are also introduced and shown
to admit solutions with singular support (e.g., peakons).Comment: 35 pages, 5 figures, 3rd version. To appear in J Nonlin Sc
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