We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one–matrix model. We define its first two Hamil-tonian structures, the recursion operator and Lax–pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new...

In this paper we discuss the dispersionless limit of the multicomponent 2D Toda hierarchy. The discrete flows of the hierarchy are used to define charge preserving Lax and Orlov–Schulman operators. This construction allows us to perform two types of dispersionless limits, one type leads to the 0-genus universal Whitham hierarchy while the other leads to a dispersionless hierarchy which contains...

We classify all the solutions to the elliptic Toda system associated to a general simple Lie algebra with singular sources at the origin and with finite energy. The solution space is shown to be parametrized by a subgroup of the corresponding complex Lie group. We also show the quantization result for the finite integrals. This work generalizes the previous works in [LWY12] and [Nie16] for Toda...

It is well-known that the Toda Theories can be obtained by reduction from the Wess-Zumino-Novikov-Witten (WZNW) model, but it is less known that this WZNW → Toda reduction is 'incomplete'. The reason for this incompleteness being that the Gauss decomposition used to define the Toda fields from the WZNW field is valid locally but not globally over the WZNW group manifold, which implies that actu...

McKean and Trubowitz [2] showed that the theory of the KdV equation ∂ ∂t g(x, t) = ∂ 3 ∂x 3 g(x, t) − 6g(x, t) ∂g ∂x (x, t). is intimately related to the geometry of a related hyperelliptic curve of infinite genus, the Bloch spectrum B g t of the operator L g t : ψ → d 2 dx 2 ψ(x) + g(x, t)ψ(x), where g t = g(x, t). As was known classically, B g t is independent of t, when g(x, t) evolves accor...

In this paper we present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. We consider both nonperiodic and periodic boundary conditions. For the nonperiodic case the phase space is the Riemann sphere, while in the periodic case it is the torus C/Z2. In both cases we prove precise estimat...

The sDiff(2) Toda equation determines all self-dual, vacuum solutions of the Einstein field equations with one rotational Killing vector. Some history of the searches for non-trivial solutions is given, including those that begin with the limit as n → ∞ of the An Toda lattice equations. That approach is applied here to the known prolongation structure for the Toda lattice, hoping to use Bäcklun...

We show that the finite Toda lattice with complex-valued initial data can be integrated by the inverse spectral method. For this goal spectral data for complex Jacobi matrices are introduced and an inverse spectral problem from the spectral data is solved. The time evolution of the spectral data for the Jacobi matrix associated with the solution of the Toda lattice is computed. Using the soluti...

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n+1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to CP. In order to compute the Leray-Schcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our ...

Using a contraction procedure, we obtain Toda theories and their structures, from affine Toda theories and their corresponding structures. By structures, we mean the equation of motion, the classical Lax pair, the boundary term for half line theories, and the quantum transfer matrix. The Lax pair and the transfer matrix so obtained, depend nontrivially on the spectral parameter. A Toda field th...