Lenard scheme for two dimensional periodic Volterra chain

Multi-soliton, multi-positon, multi-negaton, and multi-periodic solutions of the coupled Volterra lattice equation

This paper aims to find new explicit solutions including multi-soliton, multi-positon, multi-negaton, and multi-periodic for a coupled Volterra lattice system which is an integrable discrete version of the coupled KdV equation. The dynamical properties of these new solutions are discussed in detail.

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N. We derive soliton solutions of arbitrary rank $k$ and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs …

The Algebro-Geometric Initial Value Problem for the Relativistic Lotka-Volterra Hierarchy and Quasi-Periodic Solutions

We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve $\mathcal{K}_n$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\tildeϕ$ on $\mathcal{K}_p$ and trace formulas, the complex-valued algebro-geometric solutions of of RLV hierarchy are derived.

Volterra链的第一积分

First integrals for the Volterra chain

基于Volterra链特解的Wronskian公式，获得新的行列式等式，并利用Toda与Volterra链的关系，计算出Volterra链的新第一积分，进一步研究了五周期的Volterra链。

On algebro-geometric Poisson brackets for the Volterra lattice

A generalization of the theory of algebro-geometric Poisson brackets on the space of finite-gap Schroedinger operators, developped by S. P. Novikov and A. P. Veselov, to the case of periodic zero-diagonal difference operators of second order is proposed. A necessary and sufficient condition for such a bracket to be compatible with higher Volterra flows is found.

The Volterra Integrable case

In this short note we reconsider the integrable case of the Hamiltonian N-species Volterra system, as it has been introduced by Vito Volterra in 1937. In the first part, we discuss the corresponding conserved quantities, and comment about the solutions of the equations of motion. In the second part we focus our attention on the properties of the simplest model, in particular on period and frequencies of the periodic orbits. The discussion and the results presented here are to be viewed as a complement to a more general work, devoted to the construction of "a global stationary state model for a…