The ILW hierarchy

In this paper, we present a new hierarchy which includes the intermediate long wave (ILW) equation at the lowest order. This hierarchy is thought to be a novel reduction of the 1st modified KP type hierarchy. The framework of our investigation is Sato theory.Comment: 8 pages, LaTeX2

Explode-decay dromions in the non-isospectral Davey-Stewartson I (DSI) equation

In this letter, we report the existence of a novel type of explode-decay dromions, which are exponentially localized coherent structures whose amplitude varies with time, through Hirota method for a nonisospectral Davey-Stewartson equation I discussed recently by Jiang. Using suitable transformations, we also point out such solutions also exist for the isospectral Davey-Stewartson I equation itself for a careful choice of the potentials

Backlund transformation and special solutions for Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations

Using the Weiss method of truncated singular expansions, we construct an explicit Backlund transformation of the Drinfeld-Sokolov-Satsuma-Hirota system into itself. Then we find all the special solutions generated by this transformation from the trivial zero solution of this system.Comment: LaTeX, 5 page

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

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

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