From Toda to KdV

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty$ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm$. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$

Cnoidal Waves on Fermi-Pasta-Ulam Lattices

We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schr\"odinger operator with KdV cnoidal wave potential.Comment: 25 pages, 3 figure

Magnetization in the zig-zag layered Ising model and orthogonal polynomials

We discuss the magnetization $M_m$ in the $m$-th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff-Ceva fermions and orthogonal polynomials techniques. Our main result gives an explicit representation of $M_m$ via $m\times m$ Hankel determinants constructed from the spectral measure of a certain Jacobi matrix which encodes the interaction parameters between the columns. We also illustrate our approach by giving short proofs of the classical Kaufman-Onsager-Yang and McCoy-Wu theorems in the homogeneous setup and expressing $M_m$ as a Toeplitz+Hankel determinant for the homogeneous sub-critical model in presence of a boundary magnetic field.Comment: minor updates + Section 5.3 added; 38 page

Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited

We investigate soliton solutions of the Toda hierarchy on a quasi-periodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton on a quasi-periodic finite-gap background. Furthermore, we consider short range perturbations via scattering theory. We give a full description of the effect of the double commutation method on the scattering data and establish the inverse scattering transform in this setting.Comment: 16 page

A Critical Phenomenon in Solitonic Ising Chains

We discuss a phase transition of the second order taking place in non-local 1D Ising chains generated by specific infinite soliton solutions of the KdV and BKP equations.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Resonant Geometric Phases for Soliton Equations

The goal of the present paper is to introduce a multidimensional generalization of asymptotic reduction given in a paper by Alber and Marsden [1992], to use this to obtain a new class of solutions that we call resonant solitons, and to study the corresponding geometric phases. The term "resonant solitons" is used because those solutions correspond to a spectrum with multiple points, and they also represent a dividing solution between two different types of solitons. In this sense, these new solutions are degenerate and, as such, will be considered as singular points in the moduli space of solitons

An integrable 3D lattice model with positive Boltzmann weights

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0<q<1 and three continuous field variables. The layer-to-layer transfer matrices of the model form a two-parameter commutative family. This is the first example of a solvable 3D lattice model with non-negative Boltzmann weights.Comment: HyperTex is disabled due to conflicts with some macro

Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $\beta \to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.Comment: 13 figure