Toda System Research Articles

Perfectly invisible mathcal{P}mathcal{T} -symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry

We investigate a special class of the mathcal{P}mathcal{T} -symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the mathcal{P}mathcal{T} -regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the mathcal{P}mathcal{T} -regularized kinks arising as traveling waves in the field-theoretical Liouville and SU(3) conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional mathcal{N}=2 supersymmetry is extended here to the mathcal{N}=4 nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.

Phase portraits of the full symmetric Toda systems on rank-2 groups

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,ℝ) and the real form of G2.

On a family of KP multi–line solitons associated to rational degenerations of real hyperelliptic curves and to the finite non–periodic Toda hierarchy

We classify the soliton data in the totally non--negative part of Gr(k,n) which may be associated to algebraic-geometric data on certain rational degenerations of regular hyperelliptic M-curves. Such degenerate rational curve G is a desingularization of that constructed in Abenda-Grinevich (arXiv:1506.00563) for soliton data in the totally positive part of Gr(n-1,n) and the KP wavefunctions are the same in such case. G is also the curve constructed in the paper for soliton data in the totally positive part of Gr(1,n). For any such G and for any fixed k between 2 and (n-2), we show that k-compatible soliton data correspond to a family of KP multi-line solitons (T-hyperelliptic) which parametrize soliton data in an (n-1)--dimensional variety of the totally positive part of the Grassmannian Gr(k,n). We explicitly characterize T-hyperelliptic solitons from the algebraic-point of view. T-hyperelliptic solitons are also connected to the solutions of the finite non-periodic Toda hierarchy because the tau function is the same for both systems. We investigate such relation from the algebraic point of view and compare the two spectral problems. In particular, the vacuum KP divisor and the Toda divisor coincide, while k--compatible divisors may be recursively constructed using known recursions for the Toda system. Finally, we also explain how KP divisors change under the space--time transformation which induces a duality tranformation from soliton data in Gr(k,n) to soliton data in Gr(n-k,n) and compare the action of such transformation both for the KP and the Toda systems.

Variational analysis of Toda systems

The author surveys some recent progress on the Toda system on a twodimensional surface Σ, arising in models from self-dual non-abelian Chern-Simons vortices, as well as in differential geometry. In particular, its variational structure is analysed, and the role of the topological join of the barycentric sets of Σ is shown.

Geometric Quantization of Finite Toda Systems and Coherent States

Adler had showed that the Toda system can be given a coadjoint 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 positive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module. We find the Rawnsley coherent states after completion of the above space of sections. We also find non-unitary finite dimensional quantum Hilbert spaces for the system. Finally we give an expression for the quantum Hamiltonian for the system.

B 2 and G2 Toda systems on compact surfaces: A variational approach

We consider the B2 and G2 Toda systems on a compact surface (Σ, g), namely, systems of two Liouville-type PDEs coupled with a matrix of coefficients A=(aij)=2−1−22 or 2−1−32. We attack the problem using variational techniques, following the previous work [Battaglia, L. et al., Adv. Math. 285, 937–979 (2015)] concerning the A2 Toda system, namely, the case A=2−1−12. We get the existence and multiplicity of solutions as long as χ(Σ) ≤ 0 and a generic choice of the parameters. We also extend some of the results to the case of general systems.

Min–max schemes for SU(3) Toda systems

This paper surveys some recent results on Toda systems of Liouville equations. These systems model self-dual non-abelian Chern–Simons vortices, and arise in the study of holomorphic curves. Suitable min–max schemes are employed, leading to existence of solutions in several situations. We will use in particular properties about concentration of exponential functions in order to describe low-energy levels of the Euler–Lagrange energy.

Toda Systems, Cluster Characters, and Spectral Networks

We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on SLn-character varieties for n > 2 where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the SL2 case.

Local profile of fully bubbling solutions to $\mathrm {SU} (n+1)$ Toda systems

In this article we prove that for locally defined singular SU(n+1) Toda systems in R 2 , the profile of fully bubbling solutions near the singular so urce can be accurately approximated by global solutions. The main ingredients of our new approach are the classification theorem of Lin-Wei-Ye [22] and the non -degeneracy of the linearized Toda system [22], which make us overcome the difficul ties that come from lack of symmetry and the singular source.

On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

Motivated by the study of non abelian Chern Simons vortices of non topological type in Gauge Field Theory, we analyse the solvability of planar Liouville systems of Toda type in presence of singular sources. We identify necessary and sufficient conditions on the "flux" pair which ensure the radial solvability of the system. Since the given system includes the (integrable) 2 X 2 Toda system as a particular case, thus we recover the existence result available in this case. Our method relies on a blow-up analysis, which even in the radial setting, takes new turns compared with the single equation case.

Classification of solutions to Toda systems of types C and B with singular sources

In this paper, the classification in Lin et al. (Invent. Math. 190(1):169–207, 2012) of solutions to Toda systems of type A with singular sources is generalized to Toda systems of types C and B. Like in the A case, the solution space is shown to be parametrized by the abelian subgroup and a subgroup of the nilpotent subgroup in the Iwasawa decomposition of the corresponding complex simple Lie group. The method is by studying the Toda systems of types C and B as reductions of Toda systems of type A with symmetries. The theories of Toda systems as integrable systems as developed in Leznov (Teoret. Mat. Fiz. 42(3):343–349, 1980), Leznov and Saveliev (Group-theoretical methods for integration of nonlinear dynamical systems, Progress in Physics, vol. 15. Birkhauser, Verlag, 1992), Nie (J. Geom. Phys. 62(12):2424–2442, 2012), Nie (J. Nonlinear Math. Phys. 21(1):120–131, 2014), in particular the W-symmetries and the iterated integral solutions, play essential roles in this work, together with certain characterizing properties of minors of symplectic and orthogonal matrices.

Constrained lattice-field hierarchies and Toda system with Block symmetry

In this paper, we construct the additional [Formula: see text]-symmetry and ghost symmetry of two-lattice field integrable hierarchies. Using the symmetry constraint, we construct constrained two-lattice integrable systems which contain several new integrable difference equations. Under a further reduction, the constrained two-lattice integrable systems can be combined into one single integrable system, namely the well-known one-dimensional original Toda hierarchy. We prove that the one-dimensional original Toda hierarchy has a nice Block Lie symmetry.

Asymmetric blow-up for the SU(3) Toda system

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

We consider the SU(3) singular Toda system on a compact surface (Σ,g){−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑m=1Mα1m(δpm−1)−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑m=1Mα2m(δpm−1), where hi are smooth positive functions on Σ, ρi∈R+, pm∈Σ and αim>−1.We give both existence and non-existence results under some conditions on the parameters ρi and αim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities.

New blow-up phenomena for SU(n + 1) Toda system

We consider the SU(n+1) Toda system(Sλ){Δu1+2λeu1−λeu2−⋯−λeuk=0inΩ,Δu2−λeu1+2λeu2−⋯−λeuk=0inΩ,⋮⋱⋮Δuk−λeu1−λeu2−⋯+2λeuk=0inΩ,u1=u2=⋯=uk=0on∂Ω. If 0∈Ω and Ω is symmetric with respect to the origin, we construct a family of solutions (u1λ,…,ukλ) to (Sλ) such that the i-th component uiλ blows-up at the origin with a mass 2i+1π as λ goes to zero.

A topological join construction and the Toda system on compact surfaces of arbitrary genus

We consider a Toda system of Liouville equations defined on a compact surface which arises as a model for non-abelian Chern-Simons vortices. For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in \mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components.

Cluster characters and the combinatorics of Toda systems

We survey some connections between Toda systems and cluster algebras. One of these connections is based on representation theory: it is known that Laurent expansions of cluster variables are generating functions of Euler characteristics of quiver Grassmannians, and the same turns out to be true of the Hamiltonians of the open relativistic Toda chain. Another connection is geometric: the closed nonrelativistic Toda chain can be regarded as a meromorphic Hitchin system and studied from the standpoint of spectral networks. From this standpoint, the combinatorial formulas for the Hamiltonians of the open relativistic system are sums of trajectories of differential equations defined by the closed nonrelativistic spectral curves.

Cluster characters and the combinatorics of Toda systems

Исследованы некоторые связи между системами Тоды и кластерными алгебрами. Одна из этих связей основана на теории представлений: известно, что разложения Лорана кластерных переменных являются производящими функциями характеристик Эйлера грассманианов колчана, то же самое оказывается верным и для гамильтонианов открытой релятивистской цепочки Тоды. Другая связь геометрическая: замкнутую нерелятивистскую цепочку Тоды можно считать мероморфной системой Хитчина и изучать с точки зрения спектральных сетей. С этой точки зрения комбинаторные формулы для гамильтонианов открытой релятивистской системы представляют собой суммы траекторий дифференциальных уравнений, заданных с помощью замкнутых нерелятивистских спектральных кривых.

Energy Concentration and A Priori Estimates forB2andG2Types of Toda Systems

Energy Concentration and A Priori Estimates for<i>B</i>₂and<i>G</i>₂Types of Toda Systems

Type II ancient compact solutions to the Yamabe flow

Abstract We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t → - ∞ {t\to{-}\infty} , to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments, based on sharp estimates on ancient solutions of the approximated linear equation and careful estimation of the error terms which allow us to make the right choice of parameters. Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result generalizes to the gluing of k spheres for any k ≥ 2 {k\geq 2} , in such a way the configuration of radii of the spheres glued is driven as t → - ∞ {t\to{-}\infty} by a First order Toda system.