Recent Developments in Integrable Systems and Riemann-Hilbert Problems

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Abstract In this paper a theory is developed for obtaining families of solutions to the KdV equation by formulating a Riemann—Hilbert problem with an appropriate shift.

Project Overview - IPaDEGAN

The theory developed in this paper lends itself easily to obtaining explicit solutions. Examples where the subspace W can be associated to soliton type solutions are considered. More complex systems where singularities and Riemann surfaces play a role are also presented.

The computational theory of Riemann–Hilbert problems (Lecture 1) by Thomas Trogdon

The theory developed in this paper can easily be applied to other integrable systems and, eventually, to discrete integrable systems. Authors Close.

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Abstract: The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow.

We consider the problem of integrability on one energy level. This problem can be reduced to a remarkable Semi-hamiltonian system of quasilinear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral.

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Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian quasilinear system of PDEs. Abstract: In this report, I will give a rigorous mathematical construction of the Fukaya category of Laudau-Ginzburg model, which were described briefly by recent work of Gaiotto-Moore-Witten and Kapranov-Kontsevich-Soibelman on the algebra of the infraed.

This is a moduli problem about Witten equation with Lefschetz boundary condition. This is a joint work with Dingyu Yang and Wenfeng Jiang. Abstract: There are two ways elliptic curves can play a role in integrable systems: either as elliptic type solutions i. Then, I will introduce a method used in DIS, Cauchy matrix approach, which is based on the Sylvester equation and discrete dispersion relations.

Next, we show that the Cauchy matrix approach works for the study of some elliptic Integrable systems, i. Abstract: I'll talk about several results on commuting ordinary differential operators of rank two with polynomial coefficients, obtained in joint works together with A. Mironov and with I. These results are related with the following Berest conjecture. A conjecture proposed by Yu. Abstract: In this talk, we will discuss the Poisson structure on Fukaya categories and Lie bialgebra structures appearing on both the cyclic cohomology of Fukaya category and the linearized contact homology of exact symplectic manifolds with contact type boundary.

Workshop on Integrable Systems and Gromov-Witten Invariants

These are joint works with X. Chen, S. Sun and X. We will further discuss the analytic difficulty of establishing a relating map between the cyclic cohomology and the linearized contact homology, a method based on virtual neighborhood techniques will be sketched. Abstract: In this talk we consider a new dispersive integrable systemof Camassa--Holm type, which possesses two distinguish limits: Long wave dispersionless limit to diagonalisable quasilinear system of first order and short wave high frequency limit to nondiagonalisable quasilinear system of first order known as the WDVV associativity equations.

Several discretizations of Riemann surfaces exist, e. Project A01 aims at developing a comprehensive theory including discrete versions of theorems such as uniformization, convergence issues and connections to mathematical physics. In recent years, an exhaustive theory has been developed to understand and construct discrete minimal surfaces. We aim to produce something similar for the construction and classification of discrete surfaces with constant mean curvature cmc. This project is based on the observation that combinatorial and geometric features of polytopes are interlocked in many different, conceptually independent, ways.

Two geometries can be considered equivalent if there exists an angle preserving transformation between them; this is a so called conformal transformation. In the smooth case, conformal equivalences are quite well understood.

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However, mimicking their construction in the discrete case brought up not only interesting properties and algorithms, but also interesting problems - first and foremost the question of how to construct conformal deformations with certain prescribed properties. A completely analogous construction associates to each punctured Riemann surface a polyhedral fan, whose cones correspond to the ideal tessellations of the surface that occur as horocyclic Delaunay tessellations in the sense of Penner's convex hull construction.

We suggest to call this fan the secondary fan of the punctured Riemann surface.

SIAM Journal on Mathematical Analysis

The purpose of this project is to study these secondary fans of Riemann surfaces and explore how their geometric and combinatorial structure can be used to answer questions about Riemann surfaces, algebraic curves, and moduli spaces. Ropelength is a mathematical model of tying a knot or link tight in real rope: we minimize the length of a curve while keeping a unit-diameter tube around the curve embedded. We have previously developed a theory of ropelength criticality; this allows explicit descriptions of critical configurations of links like the Borromean rings and the clasp.

These configurations, whose exact geometry is quite intricate, are conjectured to be minimizers, but only known to be critical. This project will move beyond three-dimensional Euclidean space to study the ropelength problem for links in a flat three-torus. These of course lift to triply periodic structures in Euclidean space, which can include both compact and noncompact components.

In recent years, there has been great interest among differential geometers for discovered discrete integrable systems, since many discrete systems with important applications turned out to be integrable. However, there is still no exhaustive classification of these systems, in particular concerning their geometric and combinatoric structure. Here, we investigate and classify multidimensional discrete integrable systems. Riemann-Hilbert Problems RHP are another way of expressing equations satisfying a special property and have some advantages over the traditional forms.

Take for example an equation describing the motion of a water wave and its current state: Both the traditional form and the RHP form of the equation enables us to calculate the state of the wave at any point in time. But with the RHP form we can accomplish this without knowing or calculating anything about the state of the wave in between. Many basic phenomena in solid mechanics like dislocations or plastic and elastic deformation are in fact discrete operations: small breakdowns of perfect crystalline order.

The goal of this project is thereofore to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization. Many evolution equations from physics, like those for diffusion of dissolved substances or for phase separation in alloys, describe processes in which a system tries to reach the minimum of its energy or the like as quickly as possible.

In this project, we discretize these evolution equations by optimizing discrete curves in a discrete energy landscape with respect to a discrete inertia tensor.