arXiv research

Toda flow explained as a porous medium equation.

Deformed σ-models linked to Ricci flow and Toda theories.

We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to show that the Toda flow on the manifold of full flags is Morse-Smal…

Flow approach solves Toda system equations.

Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.

In this paper we investigate flows on discrete curves in $\C^2$, $\CP^1$, and $\C$. A novel interpretation of the one dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves will be given.

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

Flows on (or variations of) discrete curves in $\R^2$ give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on $\R^{2N}$, which can be interpre…

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.

This paper begins with an observation that the isospectral leaves of the signed Toda lattice as well as the Toda flow itself may be constructed from the Tomei manifolds by cutting and pasting along certain chamber walls inside a polytope. It is also observed through examples that although there is some freedom in this …

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe's ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations ari…

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…

Classifies Toda-type tt*-structures and their fixed points.

Complete solutions found for Toda equations on non-compact surfaces.

Study Toda systems blowup masses linked to Weyl groups.

Proves existence and uniqueness of solutions for A_n tt*-Toda equations.

New minimal surfaces found using Toda lattice and integrable systems.

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…

In 1967, Japanese physicist Morikazu Toda published the seminal papers exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric,…

The paper classifies Toda equations for noncompact symmetric spaces and their solutions.

Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions …

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

I show that solutions of the SU(infinity) Toda field equation generating a fixed Einstein-Weyl space are governed by a linear equation on the Einstein-Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein-Weyl space are found. I also give a classification of Einstein-Weyl sp…

Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.

New method for geodesics of multivariate normals, derived from a Toda lattice.

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with…

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…

Classifies solutions of Toda equations near singularities.

Constructs positive energy representations from Toda equations Stokes data.

This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the viewpoint on the Toda field theories as Darboux integrable differential systems as…

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy identities for the spinor parts of a blow-up sequence of solutions for which there are p…

We introduce a dynamical system which we call the AdaBoost flow. The flow is defined by a system of ODEs with control. We show that three algorithms of the AdaBoost family (i) the AdaBoost algorithm of Schapire and Freund (ii) the arc-gv algorithm of Breiman (iii) the confidence rated prediction of Schapire and Singer …

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. W…

Study of symplectic groupoids from tt*-Toda equations.

In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compa…

Extends Toda system existence results to negative functions.

Paper studies existence of Toda systems with sign-changing functions.

Adler had shown in 1979 that the Toda system can be given a coad- joint orbit description. We quantize the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with pos- itive diagonal entries. We get a unitary representation of the group with square inte…

We consider the Toda systems of VHS type with singular sources and provide a criterion for the existence of solutions with prescribed asymptotic behaviour near singularities. We also prove the uniqueness of solution. Our approach uses Simpson's theory of constructing Higgs-Hermitian-Yang-Mills metrics from stability.

We prove that all immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and the k-symmetric space structure on G/T i…

Researchers solve field equations for special gravitational instantons.

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $ε$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of th…

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized…

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss t…

The objective of this paper is to present some geometric aspects of surfaces associated with theta function solutions of the periodic 2D-Toda lattice. For this purpose we identify the $(N^2-1)$-dimensional Euclidean space with the ${\frak su}(N)$ algebra which allows us to construct the generalized Weierstrass formula …

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural c…

In this paper we establish a uniform $C^{2,θ}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,θ}$ estimate …