Related papers: Toda lattice and Riemann type minimal surfaces
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…
Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a…
Based on the work of Guillarmou, Kupiainen, and Rhodes, we construct compactified imaginary Toda theory on closed Riemann surfaces, extending the rank-one construction to the higher-rank setting. This theory is expected to describe critical…
In this paper, we extend our previous work to construct (0,2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0,2) mirrors to GLSMs on…
We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this…
It is shown how to study the 2-D Toda system for SU(n+1) using Nevanlinna theory of meromorphic functions and holomorphic curves. The results generalize recent results of Jost - Wang and Chen - Li.
The negative integrable hierarchies of shallow water waves and dispersionless Toda lattice equations are considered. The integrability is shown by explicit construction of an infinite set of conservation laws.
We use the generalized Cauchy matrix approach to derive the N-soliton solutions for the (2+2)-dimensional Toda lattice.
In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper…
The direct linearisation framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional Lie algebras $A_\infty$, $B_\infty$ and $C_\infty$, as well as the Kac--Moody algebras $A_{r}^{(1)}$,…
We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and…
We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$.…
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 $\tau$-functions of the Toda lattice…
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal…
We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schr\"odinger equation or in nonlinear models in…
By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as…
Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_{\lambda}$, $\lambda\in (0,\infty)$, of periodic properly embedded minimal surfaces in $\mathbb{R}^3$ with the property that every horizontal…
Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -\Delta u_1 = 2\rho_1(h_1e^{u_1}-1) - \rho_2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho_2(h_2e^{u_2}-1) - \rho_1(h_1e^{u_1}-1),…
Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of…
We reexamine the $W_{\infty}$ symmetry of the $sl(N)$ Conformal Affine Toda theories. It is shown that it is possible to reduce (nonuniquely) the zero curvature equation to a Lax equation for a first order pseudodifferential oprator, whose…