Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space \(\mathrm{Diff }_{1}(\mathbb R)\) equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat \(L^2\) -metric. Here \(\mathrm{Diff }_{1}(\mathbb R)\) denotes the extension of the group of all compactly supported, rapidly decreasing, or \(W^{\infty ,1}\) -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter–Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa–Holm equation.     

Diffeomorphisms preserving R‐circles in three dimensional CR manifolds

R ‐circles in (non‐degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on S³ viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.     

Well-posedness of modified Camassa-Holm equations

The Camassa-Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H¹ metric. We derive the geodesic equations on that group with respect to the invariant Hk metric, which we call the modified Camassa-Holm equation, and then study the well-posedness and dynamics of a modified Camassa-Holm equation on the unit circle S, which has some significant difference from that of Camassa-Holm equation, e.g., it does not admit finite time blowup solutions.     

Low-Regularity Solutions of the Periodic Camassa–Holm Equation

We prove local existence and uniqueness of weak solutions of the Camassa–Holm equation with periodic boundary conditions in various spaces of low-regularity which include the periodic peakons. The proof uses the connection of the Camassa–Holm equation with the geodesic flow on the diffeomorphism group of the circle with respect to the L ² metric.     

On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ^(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k = 0,1, each of them defines a smooth Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg–de Vries (KdV) equation (k = 0) is not a local diffeomorphism near the origin. A. Constantin: Supported in part by the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). T. Kappeler: Supported in part by the SNSF, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652)     

Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms

We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.     

Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein geometry of the space of densities.     

Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein geometry of the space of densities. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

On a problem of S.L. Sobolev

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.     

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

Analysis of the one-dimensional Euler–Lagrange equation of continuum mechanics with a Lagrangian of a special form

The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed.One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.     

Topological rearrangement of a function

We extend our results on weak diffeomorphism classes and decompositions of Sobolev functions to a more general framework. We introduce a family of decompositions of Sobolev functions W₀^1,p rich enough that we conjecture it allows decomposition of all Sobolev functions, not just the “craterless” ones considered in [7]. The associated weak diffeomorphism classes of a W₀^1,p Sobolev function are weakly closed when p ≥ n.     

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.     

Minimal Lagrangian surfaces in {\mathbb {CH}^2}and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of ?i?eica (or Toda) type to construct a minimal Lagrangian surface in ${\mathbb {CH}^2}$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the ${\mathbb {R}}$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the ?i?eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.     

Geodesic completeness for Sobolev H s -metrics on the diffeomorphism group of the circle

We prove that the weak Riemannian metric induced by the fractional Sobolev norm H ^s on the diffeomorphism group of the circle is geodesically complete, provided that s > 3/2.     

Dynamical models of molecular chains and efficient integration algorithms

Chains of point masses and chains of rigid bodies are used to model biological polymers. To investigate their dynamics we propose a method which allows an efficient realization of the constraints jointly with a simple and accurate integration of the free rigid body motion. The method is quite effective to evolute the geodesic flow of a rigid body chain and the global performance depends on the computational complexity of the algorithms used to compute the interaction forces. Our approach is suitable to describe a chain of rigid bodies immersed in a thermal bath. In the method we propose, the constraints are realized by hard springs whose elastic constant is set to maximize the energy dissipation rate of a Runge–Kutta integrator scheme. Moreover the use of local Lagrangian coordinates is introduced using the possibility of a continuous change of chart, such that the distance from the coordinate singularities is the highest possible. For a chain of point masses the numerical results are checked with another method where the constraints are exactly realized by means of Lagrangian coordinates. When the chain is subject to regular interactions potentials plus a thermal bath the exact and approximate constraints realization provide comparable results.     

An algebraic theory for generalized Jordan chains and partial signatures in the Lagrangian Grassmannian

We discuss an algebraic theory for generalized Jordan chains and partial signatures, that are invariants associated to sequences of symmetric bilinear forms on a vector space. We introduce an intrinsic notion of partial signatures in the Lagrangian Grassmannian of a symplectic space that does not use local coordinates, and we give a formula for the Maslov index of arbitrary real analytic paths in terms of partial signatures.     

Validity of the Korteweg–de Vries approximation for the two‐dimensional water wave problem in the arc length formulation

We consider the two‐dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. It has been proven by several authors that long‐wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the Korteweg–de Vries equation or, for certain values of the surface tension, by solutions of the Kawahara equation. These proofs are formulated either in Lagrangian or in Eulerian coordinates. In this paper, we provide a new proof, which is simpler, more elementary, and shorter. Moreover, the rigorous justification of the KdV approximation can be given for the cases with and without surface tension together by one proof. In our proof, we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties. This formulation of the water wave problem has already been of great usefulness for Ambrose and Masmoudi to simplify the proof of the local well‐posedness of the water wave problem in Sobolev spaces. © 2011 Wiley Periodicals, Inc.     

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincaré reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Now we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant with respect to left translations on this group, and assume that the mass geometry f the system may change under the action of internal control forces. Such a system can also be reduced to a Lie group. Without controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and, therefore, its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.