Yuri N. Fedorov

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

Abstract: An extension of the algebraic-geometric method for nonlinear integrable PDEs is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDEs, which are associated to nonlinear subvarieties of hyperelliptic Jacobians.The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDEs that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDEs to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations.Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

Abstract We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere Sn-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).

Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

Algebraic geometrical solutions of a new shallow-water equation and Dym-type equation are studied in connection with Hamiltonian flows on nonlinear subvarieties of hyperelliptic Jacobians. These equations belong to a class of N-component integrable systems generated by Lax equations with energy-dependent Schrodinger operators having poles in the spectral parameter. The classes of quasi-periodic and soliton-type solutions of these equations are described in terms of theta- and tau-functions by using new parametrizations. A qualitative description of real-valued solutions is provided.

Discrete nonholonomic LL systems on Lie groups

This paper studies discrete nonholonomic mechanical systems whose configuration space is a Lie group G. Assuming that the discrete Lagrangian and constraints are left-invariant, the discrete Euler–Lagrange equations are reduced to the discrete Euler–Poincare–Suslov equations. The dynamics associated with the discrete Euler–Poincare–Suslov equations is shown to evolve on a subvariety of the Lie group G. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and discrete momentum conservation is discussed.

On billiard solutions of nonlinear PDEs

This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and finite dimensional billiard systems moving inside n-dimensional quadrics under the field of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.

Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

The algebraic-geometric approach is extended to study evolution equations associated with the energy-dependent Schr?dinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvarieties of Jacobi varieties. The general approach is demonstrated by using new parametrizations for constructing quasi-periodic solutions of the shallow-water and Dym-type equations in terms of theta-functions. A qualitative description of real-valued solutions is provided.

Homoclinic billiard orbits inside symmetrically perturbed ellipsoids

The billiard motion inside an ellipsoid of R 3 is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geometrical, dynamical and topological point of view. It contains eight surfaces, called separatrices. The splitting of the separatrices under symmetric perturbations of the ellipsoid is studied using a symplectic discrete version of the Poincar´ e– Melnikov method, with special emphasis on the following situations: close to the flat limit (when the minor axis of the ellipsoid is small enough), close to the oblate limit (when the ellipsoid is close to an ellipsoid of revolution around its minor axis) and close to the prolate limit (when the ellipsoid is close to an ellipsoid of revolution around its major axis). It is proved that any non-quadratic entire symmetric perturbation breaks the integrability and splits the separatrices, although (at least) 16 symmetric homoclinic orbits persist. Close to the flat limit, these orbits become transverse under very general polynomial perturbations of the ellipsoid. Finally, a particular quartic symmetric perturbation is analysed in great detail. Close to the flat and to the oblate limits, the 16 symmetric homoclinic orbits are the unique primary homoclinic orbits. Close to the prolate limit, the number of primary homoclinic orbits undergoes infinitely many bifurcations. The first bifurcation curves are computed numerically. The planar and high-dimensional cases are also discussed.

Quasi-Chaplygin Systems and Nonholonimic Rigid Body Dynamics

We show that the Suslov nonholonomic rigid body problem studied in by Fedorov and Kozlov (Am. Math. Soc. Transl. Ser. 2 168:141–171, 1995), Jovanović (Reg. Chaot. Dyn. 8(1):125–132, 2005), and Zenkov and Bloch (J. Geom. Phys. 34(2):121–136, 2000) can be regarded almost everywhere as a generalized Chaplygin system. Furthermore, this provides a new example of a multidimensional nonholonomic system which can be reduced to a Hamiltonian form by means of Chaplygin reducing multiplier. Since we deal with Chaplygin systems in the local sense, the invariant manifolds of the integrable examples are not necessary tori

Closed Geodesics and Billiards on Quadrics Related to Elliptic KdV Solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in