Algebraic integrability of confluent Neumann system

In this paper we study the Neumann system, which describes the harmonic oscillator (of arbitrary dimension) constrained to the sphere. In particular we will consider the confluent case where two eigenvalues of the potential coincide, which implies that the system has S^{1} symmetry. We will prove complete algebraic integrability of confluent Neumann system and show that its flow can be linearized on the generalized Jacobian torus of some singular algebraic curve. The symplectic reduction of S^{1} action will be described and we will show that the general Rosochatius system is a symplectic quotient of the confluent Neumann system, where all the eigenvalues of the potential are double. This will give a new mechanical interpretation of the Rosochatius system.Comment: 17 pages, 1 figur

5-dimensional contact SO(3)-manifolds and Dehn twists

In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifoldswith singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists.Comment: 16 pages, 1 figure; simplification of arguments by restricting classification to coorientation preserving contactomorphism

Billiard algebra, integrable line congruences, and double reflection nets

The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.Comment: 18 pages, 8 figure

Period Integrals of CY and General Type Complete Intersections

We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.Comment: An erratum is included to correct Theorem 3.12 (Uniqueness of CY structure

Paleoseismology of a major crustal seismogenic source near Mexico City. The southern border of the Acambay Graben

The Trans-Mexican Volcanic Belt is an active continental volcanic arc related to subduction along the Middle America trench. It is characterized by intra-arc extension resulting into several major arc-parallel active fault systems and tectonic basins. The Acambay graben, one of the largest of these basins, is located near Mexico City, in the central part of this province. In 1912, a M 6.9 earthquake ruptured the surface along the northern border of the graben together with at least two other faults. In this paper, we analyze the paleoseismic history of the southern border of the Acambay Graben, with new observations made in one natural outcrop and four paleoseismological trenches excavated across branches of the Venta de Bravo Fault at the site where it overlaps with the Pastores Fault. We present evidence of at least two paleo-earthquakes that occurred between 12,190 +/- 175 and 5,822 +/- 87 cal year BP and between 647 +/- 77 and 250 cal year BP. On one of these branches, we estimate a minimum slip-rate value between 0.1 and 0.23 mm/year for the last 12 ka and a mean recurrence interval of 8.5 +/- 3 ka. By considering several likely rupture lengths along the Venta de Bravo and Pastores faults, we calculated a maximum possible magnitude of M-w 7.01 +/- 0.27. Finally, by correlating events recorded along different faults within the Acambay Graben, we discuss several possible rupture coalescent scenarios and related consequences for Mexico City

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\R\ltimes \R^{n-1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of $n$ contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories

The Hamiltonian Structure of the Second Painleve Hierarchy

In this paper we study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable $z$ depending on n parameters denoted by ${t}_1,...,{t}_{n-1}$ and $\alpha_n$. We introduce new canonical coordinates and obtain Hamiltonians for the $z$ and $t_1,...,t_{n-1}$ evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates

Source model of the 2007 M_w 8.0 Pisco, Peru earthquake: Implications for seismogenic behavior of subduction megathrusts

We use Interferometric Synthetic Aperture Radar, teleseismic body waves, tsunami waveforms recorded by tsunameters, field observations of coastal uplift, subsidence, and runup to develop and test a refined model of the spatiotemporal history of slip during the M_w 8.0 Pisco earthquake of 15 August 2007. Our preferred solution shows two distinct patches of high slip. One patch is located near the epicenter while another larger patch ruptured 60 km further south, at the latitude of the Paracas peninsula. Slip on the second patch started 60 s after slip initiated on the first patch. We observed a remarkable anticorrelation between the coseismic slip distribution and the aftershock distribution determined from the Peruvian seismic network. The proposed source model is compatible with regional runup measurements and open ocean tsunami records. From the latter data set, we identified the 12 min timing error of the tsunami forecast system as being due to a mislocation of the source, caused by the use of only one tsunameter located in a nonoptimal azimuth. The comparison of our source model with the tsunami observations validate that the rupture did not extend to the trench and confirms that the Pisco event is not a tsunami earthquake despite its low apparent rupture velocity (<1.5 km/s). We favor the interpretation that the earthquake consists of two subevents, each with a conventional rupture velocity (2–4 km/s). The delay between the two subevents might reflect the time for the second shock to nucleate or, alternatively, the time it took for afterslip to increase the stress level on the second asperity to a level necessary for static triggering. The source model predicts uplift offshore and subsidence on land with the pivot line following closely the coastline. This pattern is consistent with our observation of very small vertical displacement along the shoreline when we visited the epicentral area in the days following the event. This earthquake represents, to our knowledge, one of the best examples of a link between the geomorphology of the coastline and the pattern of surface deformation induced by large interplate ruptures

Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear relations amongst the gradients of the integrals of motion, coincide with points where there are k (doubly) degenerate eigenvalues of representatives L and Lbar of the two inequivalent classes of Lax matrices (corresponding to degenerate periodic or antiperiodic solutions of the associated second-order difference equation). The singularities are shown to be nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold. Sigma_k is shown to be of elliptic type, and the frequencies of transverse oscillations under Hamiltonians which fix Sigma_k are computed in terms of spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is given by the product of the holonomies (equal to +/- 1) of the even- (or odd-) indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio

Extremal Sasakian Geometry on T2×S3T^2\times S^3 and Related Manifolds

We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on $T^2\times S^3$ and certain related manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.Comment: 35 pages, clarifications made and typos corrected in revised version, further corrections made, slight change in title, to appear in Compositio Mathematic