arXiv research

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodi…

Study on hypermaps and KP hierarchy, proving tau function and enumerative meaning.

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseud…

New diffiety theory leads to a well-posed KP hierarchy.

Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable syst…

We show that the quantum field theoretical formulation of the $τ$-function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic ver…

We find contact integrable extensions and coverings for the r-th double modified dispersionless Kadomtsev--Petviashvili equation.

The algebraic and geometric properties of a novel generalization of Clifford's classical C4 point-circle configuration are analysed. A connection with the integrable quaternionic discrete Schwarzian Kadomtsev-Petviashvili equation is revealed.

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $Ω_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…

We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the…

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated…

Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.

Study identifies pitfalls in assessing hierarchies for multi-class classification.

Super tau-covers extend bihamiltonian hierarchies' symmetries.

Hydrodynamic hierarchy deformed using conservation laws.

Legendre transformations link related integrable hierarchies.

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy

We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.

Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a preferred non-zero real sec…

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…

Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II

Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.

Constructs tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy

A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of $τ$-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

HAKE embeds entities in polar coordinates to model semantic hierarchies in knowledge graphs.

New integrable deformations for topological hierarchies from Frobenius manifolds.

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Co…

Paper investigates methods to improve classification by inducing a hierarchy from flat labels.

We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.

We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau fu…

Paper addresses limitations of traditional hierarchical clustering methods.

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the $CP^1$ topological sigma model and the extended Toda hierarchy. We al…

We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the…

We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebr…

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.

ProHOC detects OOD samples in class hierarchies, predicting them to correct internal nodes.

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …

The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using …

Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures.

We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorithms. In particular, w…

The paper constructs Darboux transforms for a specific hierarchy and its related flows.

Local tri-Hamiltonian structure for Ablowitz-Ladik hierarchy established.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.