Yang-Baxter maps and the discrete KP hierarchy

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples

Darboux dressing and undressing for the ultradiscrete KdV equation

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.Comment: 41 pages, 5 figures // Full, unabridged version, including two appendice

Burchnall-Chaundy polynomials and the Laurent phenomenon

The Burchnall-Chaundy polynomials Pn(z) are determined by the differential recurrence relation with The fact that this recurrence relation has all solutions polynomial is not obvious and is similar to the integrality of Somos sequences and the Laurent phenomenon. We discuss this parallel in more detail and extend it to two difference equations and related to two different KdV-type reductions of the Hirota-Miwa and Dodgson octahedral equations. As a corollary we have a new form of the Burchnall-Chaundy polynomials in terms of the initial data , which is shown to be Laurent

Two-dimensional soliton cellular automaton of deautonomized Toda-type

A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.Comment: 11 pages, LaTeX fil

Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation

Casorati determinant solution to the non-autonomous discrete KdV equation is constructed by using the bilinear formalism. We present three different bilinear formulations which have different origins

Painleve equations from Darboux chains - Part 1: P3-P5

We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three different branches, each described by an infinite chain of equations. The Painleve equations are obtained by closing the chain periodically at the lowest nontrivial level(s). The chains provide ``symmetric forms'' for the Painleve equations, from which Hirota bilinear forms and Lax pairs are derived. In this paper (Part 1) we analyze in detail the cases P3-P5, while P6 will be studied in Part 2.Comment: 23 pages, 1 reference added + minor change

On non-QRT Mappings of the Plane

We construct 9-parameter and 13-parameter dynamical systems of the plane which map bi-quadratic curves to other bi-quadratic curves and return to the original curve after two iterations. These generalize the QRT maps which map each such curve to itself. The new families of maps include those that were found as reductions of integrable lattices

The A^{(1)}_M automata related to crystals of symmetric tensors

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra U'_q(A^{(1)}_M) is introduced. It is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(A^{(1)}_{M-1}). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete KP equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.Comment: 45 pages, latex2e, 2 figure

Tropical Krichever construction for the non-periodic box and ball system

A solution for an initial value problem of the box and ball system is constructed from a solution of the periodic box and ball system. The construction is done through a specific limiting process based on the theory of tropical geometry. This method gives a tropical analogue of the Krichever construction, which is an algebro-geometric method to construct exact solutions to integrable systems, for the non-periodic system.Comment: 13 pages, 1 figur