# Some Classes of Solutions to the Toda Lattice Hierarchy

@article{Widom1997SomeCO, title={Some Classes of Solutions to the Toda Lattice Hierarchy }, author={Harold Widom}, journal={Communications in Mathematical Physics}, year={1997}, volume={184}, pages={653-667} }

Abstract. We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the l-periodic Toda hierarchy.

## 14 Citations

Toda Soliton Limits on General Backgrounds

- Mathematics
- 1999

Abstract Starting from an arbitrary background solution of the Toda lattice, we study limits of N -soliton solutions on this given background as N tends to infinity. This yields a new class of…

Harmonic Bundles and Toda Lattices With Opposite Sign II

- Mathematics, Physics
- 2014

We study the integrable variation of twistor structure associated to any solution of the Toda lattice with opposite sign. In particular, we give a criterion when it has an integral structure. It…

Asymptotics of a Class of Operator Determinants with Application to the Cylindrical Toda Equations

- Physics, Mathematics
- 2006

In work of C. A. Tracy and the author asymptotic formulas were derived for certain operator determinants which gave solutions to the cylindrical Toda equations. Later the author considered a more…

An Integral Operator Solution to the Matrix Toda Equations

- Mathematics, Physics
- 1997

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the…

Asymptotics of a class of Fredholm determinants

- Mathematics, Physics
- 1998

In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived.…

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

- Mathematics, Physics
- 1998

t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, , have the representation
where Kk$ are integral operators. This class includes the n-periodic…

Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa I. Stokes Data

- Mathematics, Physics
- 2012

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii)…

Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II: Riemann–Hilbert Problem

- Mathematics, Physics
- 2013

In Guest et al. (arXiv:1209.2045) (part I) we computed the Stokes data for the smooth solutions of the tt*-Toda equations whose existence we had previously established by p.d.e. methods. Here we…

On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes

- Physics, Mathematics
- 1997

Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann…

New Results in $${\mathcal {N}}=2$$N=2 Theories from Non-perturbative String

- Physics, Mathematics
- 2017

We describe the magnetic phase of SU(N) $${\mathcal {N}}=2$$N=2 super-Yang–Mills theories in the self-dual $$\Omega $$Ω-background in terms of a new class of multi-cut matrix models. These arise from…

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