A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Paper connects Bäcklund transformations to nonlocal pseudosymmetries.
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.
Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 Bäcklund transformations relating two hyperbolic Monge-Ampère systems. In cases when the Bäcklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new e…
In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (…
We investigate geometric aspects of the the Bäcklund transform of principal contact element nets. A Bäcklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the Bäcklund transform and prove its co…
Backlund transformations of admissible curves in the Galilean 3-space and pseudo-Galilean 3-space and also spatial Backlund transformations of space curves in Galilean 4-space preserve the torsions under certain assumptions.
New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.
Researchers classify and characterize Bäcklund transformations for hyperbolic Monge-Ampère systems.
Researchers find explicit Bäcklund transforms for specific quadrics.
In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new …
A Backlund transformation between two hyperbolic Monge-Ampere systems may be described as a certain type of exterior differential system on a 6-dimensional manifold B. The transformation is homogeneous if the group of symmetries of the system acts transitively on B. We give a complete classification of homogeneous Back…
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…
We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund transformations in arXiv:0707.4408v2 can be constructed using symmetry reduction.
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
New definition of Bäcklund transformation for surface isometric deformation.
We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …
Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution equations with one space variable. The main concept is that of connection defin…
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank- Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a clas…
We characterize Bianchi-Bäcklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.
We begin an exploration of parametric Backlund transformations for hyperbolic Monge-Ampere systems. We compute invariants for such transformations and explore the behavior of four examples regarding their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for furthe…
We investigate basic features of Bianchi's Bäcklund transformation of quadrics to see if it can be obtained under weaker assumptions and if it can be generalized to deformations of other surfaces.
We construct a local action of the group of rational maps from to on local solutions of flows of the ZS-AKNS -hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different fact…
New harmonic maps to hyperbolic plane via Bäcklund transformation.
We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation . The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter . It is also a nonlocal symmetry…
The geometry of an admissible Bäcklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal prolongation of a --graded semi-simple Lie algebra.
In the context of the cohomological deformation theory, infinitesimal description of one-parametric families of Backlund transformations of special type including classical examples is given. It is shown that any family of such a kind evolves in the direction of a nonlocal symmetry shadow.
We present a 2x2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the membe…
Let be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU(2)-connections and we show that they can be built up from suitable Bäcklund transformations.
Geometric transformations on null curves in AdS induce KdV solutions.
We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…
We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define Bäcklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to…
The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…
Harmonic maps from $\BR^2$ or one-connected domain $Ø\subset \BR^2$ into $GL(m, \BC)$ and are treated. The GBDT version of the Bäcklund-Darboux transformation is applied to the case of the harmonic maps. A new general formula on the GBDT transformations of the Sym-Tafel immersions is derived. A class of the harm…
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…
In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of th…
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
Following I. S. Krasilshchik and A. M. Vinogradov, we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Backlund transformations. We show that, similarly to usual coverings in top…
New solutions found for elliptic sinh-Gordon and sine-Gordon equations.
Unified study of harmonic maps between pseudo-Riemannian surfaces.
Skew parallelogram nets factorize, encompassing discrete differential geometry.
The Hodge star mean curvature flow on a 3-dimensional Riemannian or pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in geometric analysis. A curve flow is integrable if the local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we sho…
Study generalizes Yang-Mills equations for special complex surfaces.
A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the …
We provide a generalization of Bianchi's triply conjugate systems containing a family of deformations of 2-dimensional quadrics together with its Bäcklund transformation to higher dimensions.
We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.
Paper formulates governing equations for membrane O surfaces.