A new two-dimensional lattice model that is "consistent around a cube"

For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is "consistent around a cube" (CAC). As a consequence of CAC one can construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted a search based on this principle and certain additional assumptions. One of those assumptions was the "tetrahedron property", which is satisfied by most known equations. We present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property. Its Lax pair is also presented and some basic properties discussed.Comment: 8 pages in LaTe

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

A Bilinear Approach to Discrete Miura Transformations

We present a systematic approach to the construction of Miura transformations for discrete Painlev\'e equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of $\tau$-functions. Elimination of $\tau$-functions from the resulting system leads to another nonlinear equation, which is a ``modified'' version of the original equation. The procedure therefore yields Miura transformations. In this letter, we illustrate this approach by reproducing previously known Miura transformations and constructing new ones.Comment: 7 pages in TeX, to appear in Phys. Letts.

Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $\tau$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te

On Completely Integrability Systems of Differential Equations

In this note we discuss the approach which was given by Wazwaz for the proof of the complete integrability to the system of nonlinear differential equations. We show that his method presented in [Wazwaz A.M. Completely integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15 (2010) 2828-2835] is incorrect.Comment: 14 pages. This paper was sent to the Communications in Nonlinear Science and Numerical Simulatio

A generalization of determinant formulas for the solutions of Painlev\'e II and XXXIV equations

A generalization of determinant formulas for the classical solutions of Painlev\'e XXXIV and Painlev\'e II equations are constructed using the technique of Darboux transformation and Hirota's bilinear formalism. It is shown that the solutions admit determinant formulas even for the transcendental case.Comment: 20 pages, LaTeX 2.09(IOP style), submitted to J. Phys.

Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation

We consider in detail the self-trapping of a soliton from a wave pulse that passes from a defocussing region into a focussing one in a spatially inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger equation in which the dispersion coefficient changes its sign from normal to anomalous. The model has direct applications to dispersion-decreasing nonlinear optical fibers, and to natural waveguides for internal waves in the ocean. It is found that, depending on the (conserved) energy and (nonconserved) mass of the initial pulse, four qualitatively different outcomes of the pulse transformation are possible: decay into radiation; self-trapping into a single soliton; formation of a breather; and formation of a pair of counterpropagating solitons. A corresponding chart is drawn on a parametric plane, which demonstrates some unexpected features. In particular, it is found that any kind of soliton(s) (including the breather and counterpropagating pair) eventually decays into pure radiation with the increase of the energy, the initial mass being kept constant. It is also noteworthy that a virtually direct transition from a single soliton into a pair of symmetric counterpropagating ones seems possible. An explanation for these features is proposed. In two cases when analytical approximations apply, viz., a simple perturbation theory for broad initial pulses, or the variational approximation for narrow ones, comparison with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres

Bicomplexes and Integrable Models

We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are presented, including the nonlinear Schrodinger and sine-Gordon equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st

Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when $h=0$. They also provide rational solutions for a particular discretisation of $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.Comment: 23 pages, IOP style. Title changed, and connection with Umemura polynomials adde