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E. V. Ferapontov

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

At least 55 records · Page 3Linked to original sources
arXiv2003

Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d-1)n arbitrary functions of one variable. Given a PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions (if consistent).

nlin.SI↗
arXiv2003

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the `dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.

nlin.SI↗
arXiv2003

On integrability of (2+1)-dimensional quasilinear systems

A (2+1)-dimensional quasilinear system is said to be `integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.

nlin.SI↗
arXiv2003

Hydrodynamic reductions of the heavenly equation

We demonstrate that Plebański's first heavenly equation decouples in infinitely many ways into a triple of commuting (1+1)-dimensional systems of hydrodynamic type which satisfy the Egorov property. Solving these systems by the generalized hodograph method, one can construct exact solutions of the heavenly equation parametrized by arbitrary functions of a single variable. We discuss explicit examples of hydrodynamic reductions associated with the equations of one-dimensional nonlinear elasticity, linearly degenerate systems and the equations of associativity.

nlin.SI↗
arXiv2002

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants. Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulae provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by arbitrary functions of one variable.

nlin.SI↗
arXiv2002

Projective differential geometry of higher reductions of the two-dimensional Dirac equation

We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator $D_n$ of order $n$ mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multi-component analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that $n$-th order reductions coincide with the projective-geometric `Gauss-Codazzi' equations governing special classes of line congruences in the projective space $P^{2n-1}$, which is the projectivised kernel of $D_n$. In the second order case this leads to the theory of $W$-congruences in $P^3$ which belong to a linear complex, while the third order case corresponds to isotropic congruences in $P^5$. Higher reductions are compatible with odd-order flows of the Davey-Stewartson hierarchy. All these flows preserve the kernel $D_n$, thus defining nontrivial geometric evolutions of line congruences. Multi-component generalizations are also discussed. The correspondence between geometric picture and the theory of integrable systems is established; the definition of the class of reductions and all geometric objects in terms of the multicomponent KP hierarchy is presented. Generating forms for reductions of arbitrary order are constructed.

nlin.SI↗
arXiv2002

Decomposition of higher order equations of Monge-Ampere type

Given a function f(x, t), its fourth (symmetric) differential is a quartic form in dx, dt. It is well-known that any quartic form in two variables can be represented as a sum of three 4th powers of linear forms. The particular case of two 4th powers is characterized by the vanishing of a catalecticant determinant, which is a fourth order PDE for the function f. We demonstrate that this PDE decouples in a direct sum of two Monge-Ampere equations. A higher order generalization of this decomposition is also proposed.

nlin.SI↗
arXiv2001

Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator

The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the corresponding Gauss-Codazzi equations and draw parallels between this geometrical problem and the theory of compatible Poisson brackets of hydrodynamic type. It turns out that coordinate hypersurfaces of the n-orthogonal systems arising in the theory of compatible Poisson brackets of hydrodynamic type must necessarily possess deformations preserving the shape operator.

math.DG↗
arXiv2001

Systems of conservation laws of Temple class, equations of associativity and linear congruences in projective space

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronese variety and the theory of associativity equations of two-dimensional topological field theory.

math.DG↗
arXiv2000

Integrable Schrödinger operators with magnetic fields: factorisation method on curved surfaces

The factorisation method for Schrödinger operators with magnetic fields on a two-dimensional surface $M^2$ with non-trivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as Dirac magnetic monopole and Landau problem. The global geometric aspects and related spectral properties of the operators from the factorisation chains are discussed in details. We also consider the Laplace transformations on a curved surface and extend the class of Schrödinger operators with two integrable levels introduced in the flat case by S.P.Novikov and one of the authors.

math-ph↗
arXiv2000

Compatible Poisson brackets of hydrodynamic type

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (1) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor; (2) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (3) the connection of this problem with the class of surfaces in Euclidean space which possess nontrivial deformations preserving the Weingarten operator.

math.DG↗
arXiv1999

Transformations of quasilinear systems originating from the projective theory of congruences

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the Lévy transformations of semihamiltonian systems. Correspondence between commuting flows and certain families of planes containing the lines of the congruence is pointed out. In the particular case n=2 this construction provides an explicit parametrization of surfaces, harmonic to a given congruence. Adjoint Lévy transformations of semihamiltonian systems are discussed. Explicit formulae for the Lévy and adjoint Lévy transformations of the characteristic velocities are set down. A closely related construction of the Ribaucour congruences of spheres is discussed in the Appendix.

math.DG↗
arXiv1999

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.

math.DG↗
arXiv1998

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.

math.DG↗
arXiv1998

Surfaces with flat normal bundle: an explicit construction

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in the affine space. An explicit parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences and surfaces with flat normal bundle in the Lie quadric this gives an explicit representation of surfaces with flat normal bundle in the 4-dimensional space form of the Lorentzian signature. Direct projective analog of this construction is the known parametrization of W-congruences by three solutions of the Moutard equation. Under the Plücker embedding W-congruences give rise to surfaces with flat normal bundle in the Plücker quadric. Integrable evolutions of surfaces with flat normal bundle and parallels with the theory of nonlocal Hamiltonian operators of hydrodynamic type are discussed in the conclusion.

math.DG↗