arXiv subjects

E. V. Ferapontov

Publications and source records attributed to E. V. Ferapontov.

57 records · Page 4Linked to original sources

arXiv1997

Surfaces in Lie sphere geometry and the stationary Davey-Stewartson hierarchy

We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov equation, whose role in the theory of these surfaces is similar to that of Calapso's equation in the theory of isothermic surfaces. Since Calapso's equation itself turns out to be related to the stationary Davey-Stewartson equation, these results shed some new light on differential geometry of the stationary Davey-Stewartson hierarchy. Diagonally-cyclidic surfaces are the natural Lie sphere analogs of the isothermally-asymptotic surfaces in projective differential geometry for which we also derive the stationary modified Veselov-Novikov equation with the different real reduction. Parallels between invariants of surfaces in Lie sphere geometry and reciprocal invariants of hydrodynamic type systems are drawn in the conclusion.

dg-ga↗

arXiv1997

Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences

The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems. We discuss periodic Laplace sequences of with the emphasize on the simplest nontrivial case of period 2. For 3-component systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn to be related to a special quadratic reduction of the (2+1)-dimensional 3-wave system which can be reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2+1)-dimensional n-wave system. The 2-component Laplace transformations can be interpreted also as the symmetries of the (2+1)-dimensional integrable equations of Davey-Stewartson type. Laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.

solv-int↗

arXiv1994

On the Monge-Ampere equivalent of the sine-Gordon equation

Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit form of the correspondence between these two integrable non-linear partial differential equations using their well-known properties in differential geometry. We find that the cotangent of the angle between asymptotic directions is directly related to the mean curvature of the surface. This is a Backlund-type transformation between the sine-Gordon and Monge-Ampere equations.

solv-int↗