Related papers: Solution of the generalized periodic discrete Toda…

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…

Combinatorial expressions are presented to the solutions to initial value problems of the discrete and ultradiscrete Toda molecules. For the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of…

In this short review we compare different ways to construct solutions of the periodic Toda lattice. We give two recipes that follow from the projection method and compare them with the algebra-geometric construction of Krichever.

We study the interpolation analogue of the Hermite-Pad\'e type I approximation problem. We provide its determinant solution and we write down the corresponding integrable discrete system as an admissible reduction of Hirota's discrete…

An algorithm to obtain equations between theta functions with integral characteristics evaluated at $\tau$ and $p\tau$ for $g>1$ is presented.

This paper is a continuation of arXiv:17.01.02867. We give here rigorous solution of the parametrix problem for Toda rarefaction problem and complete asymptotic analysis, justifying the asymptotics obtained in arXiv:17.01.02867.

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type $A_n$. The principal issue is the connection formulae between…

Following a prescription of \cite{4} for a solitonic specialization of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely…

We consider the Hirota equation (the discrete analog of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogs of…

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…

Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.

The concept of a fundamental solution to the time-periodic Stokes equations in dimension $n\geq 2$ is introduced. A fundamental solution is then identified and analyzed. Integrability and pointwise estimates are established.

A general Casoratian formulation is proposed for the 2D Toda lattice equation, which involves coupled eigenfunction systems. Various Casoratian type solutions are generated, through solving the resulting linear conditions and using a…

This paper is focused on quasi-periodic wave solutions of (3+1) generalized BKP equation. Because of some difficulties in calculations of N=3 periodic solutions, hardly ever has there been a study on these solutions by using Rieamann theta…

We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral…

In this paper we prove that the following delay differential equation \[ \frac{d}{dt}x(t)=rx(t)\left(1-\int_{0}^{1}x(t-s)ds\right), \] has a periodic solution of period two for $r>\frac{\pi^{2}}{2}$ (when the steady state, $x=1$, is…

Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.

A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation…

We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szeg\"{o}, and is also interpreted as a consequence of the QR…

M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23}) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and…