Related papers: Non-integrability vs. integrability in pentagram m…

The pentagram map is a discrete integrable system first introduced by Schwartz in 1992. It was proved to be intregable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to…

A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…

In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e.,…

We study the KdV and Burgers nonlinear systems and show in a consistent way that they can be mapped to each other through a strong requirement about their evolutions's flows to be connected. We expect that the established mapping between…

In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive…

One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is…

In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples…

We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further…

We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as duality and Petrie duality.

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such…

A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality…

A plane graph is called a rectangular graph if each of its edges can be oriented either horizontally or vertically, each of its interior regions is a four-sided region and all interior regions can be fitted in a rectangular enclosure. If…

We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action-angle-angle maps with degeneracy, satisfying the intersection property. Such degenerate action-angle-angle maps arise…

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy…

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of…

We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy…

The class of 2-dimensional non-integrable flat dynamical systems has a rather extensive literature with many deep results, but the methods developed for this type of problems, both the traditional approach via Teichm\"{u}ller geometry and…

The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented…