2,358 research outputs found
Invariant description of solutions of hydrodynamic type systems in hodograph space: hydrodynamic surfaces
Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as
non-parametrized submanifolds of the hodograph space. We propose an invariant
differential-geometric characterization of hydrodynamic surfaces by expressing
the curvature form of the characteristic web in terms of the reciprocal
invariants.Comment: 12 page
Integrable systems in projective differential geometry
Some of the most important classes of surfaces in projective 3-space are
reviewed: these are isothermally asymptotic surfaces, projectively applicable
surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is
demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce
to integrable systems which are quite familiar from the modern soliton theory
and coincide with the stationary flows in the Davey-Stewartson and
Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The
corresponding Lax pairs can be obtained by inserting a spectral parameter in
the equations of the Wilczynski moving frame
Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry
It is demonstrated that the stationary Veselov-Novikov (VN) and the
stationary modified Veselov-Novikov (mVN) equations describe one and the same
class of surfaces in projective differential geometry: the so-called
isothermally asymptotic surfaces, examples of which include arbitrary quadrics
and cubics, quartics of Kummer, projective transforms of affine spheres and
rotation surfaces. The stationary mVN equation arises in the Wilczynski
approach and plays the role of the projective "Gauss-Codazzi" equations, while
the stationary VN equation follows from the Lelieuvre representation of
surfaces in 3-space. This implies an explicit Backlund transformation between
the stationary VN and mVN equations which is an analog of the Miura
transformation between their (1+1)-dimensional limits.Comment: Latex, 13 page
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