P.R. Gordoa

A nonisospectral extension of the Volterra hierarchy to 2+1 dimensions

We give a new nonisospectral generalization of the Volterra lattice equation to 2+1 dimensions. We use this to construct a new nonisospectral lattice hierarchy in 2+1 dimensions, along with its underlying linear problem. Reductions yield a variety of new integrable hierarchies, including generalizations of known discrete Painleve hierarchies, all along with their corresponding linear problems. This represents an extension of previously developed techniques to the discrete case.

Second and fourth Painlevé hierarchies and Jimbo-Miwa linear problems

The relations between the different linear problems for Painleve equations is an intriguing open problem. Here we consider our previously given second and fourth Painleve hierarchies [Publ. Res. Inst. Math. Sci. (Kyoto) 37, 327–347 (2001)], and show that they could alternatively have been derived using the linear problems of Jimbo and Miwa. That is, we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth Painleve equations themselves onto those of Jimbo and Miwa.

New 2+1 dimensional nonisospectral Toda lattice hierarchy

We give a new 2+1 dimensional nonisospectral generalization of the Toda lattice hierarchy. Reductions yield a variety of new integrable hierarchies along with their underlying linear problems, including new 1+1 dimensional differential-delay hierarchies (nonisospectral and isospectral), new ordinary differential-delay hierarchies, and new discrete Painleve hierarchies. We also show that a reduction in components yields our previously obtained 2+1 dimensional nonisospectral Volterra lattice hierarchy.

Non-isospectral lattice hierarchies in 2 + 1 dimensions and generalized discrete Painleve hierarchies

Abstract In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again along with their hierarchy of underlying linear problems. We obtain a generalized discrete first Painlevé hierarchy which includes as special cases, after further summation, both the standard discrete first Painlevé hierarchy and a new extended version of the discrete thirty-fourth Painlevé hierarchy.

A method of reduction of order for discrete systems

We show how, by establishing a suitable Backlund correspondence, a reduction in order can be effected for discrete systems having a certain structure. In this way we succeed in extending to discrete systems ideas previously developed within the context of ordinary differential equations. It is also worth emphasizing that our approach represents an application of properties characteristic of completely integrable systems outside of that context, that is, to systems of discrete equations, integrable or otherwise.

A new derivation of Painlevé hierarchies

Abstract We give a new derivation of two Painleve hierarchies. This is done by extending the accelerating-wave reductions of the Korteweg–de Vries and dispersive water wave equations to their respective hierarchies. We also consider the extension of this reduction of Burgers equation to the Burgers hierarchy.

Nonisospectral scattering problems and similarity reductions

Abstract We give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems. Special cases of these changes of variables involve similarity variables, and similarity solutions of a hierarchy are seen to correspond to time-independent solutions of an equivalent hierarchy. We thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same Painleve hierarchies. As examples we consider the Korteweg–de Vries hierarchy and the dispersive water wave hierarchy.

Auto-Bäcklund transformations and integrability of ordinary and partial differential equations

We introduce a partial differential equation whose structure, in many respects, mimics that of the second Painleve equation. For this equation we give auto-Backlund transformations of both ordinary differential equation and partial differential equation types. Our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto-Backlund transformations and the integrability of ordinary and partial differential equations.

Auto-Bäcklund transformations for a differential-delay equation

Discrete Painleve equations have, over recent years, generated much interest. One property of such equations that is considered to be particularly important is the existence of auto-Backlund transformations, that is, mappings between solutions of the equation in question, usually involving changes in the values of parameters appearing as coefficients. We have recently presented extensions of discrete Painleve equations to equations involving derivatives as well as shifts in the independent variable. Here we show how auto-Backlund transformations can also be constructed for such differential-delay equations. We emphasise that this is the first time that an auto-Backlund transformation has been given for a differential-delay equation.

The Prelle-Singer method and Painlevé hierarchies

We consider systems of ordinary differential equations (ODEs) of the form BK=0, where B is a Hamiltonian operator of a completely integrable partial differential equation hierarchy, and K = (K, L)T. Such systems, while of quite low order and linear in the components of K, may represent higher-order nonlinear systems if we make a choice of K in terms of the coefficient functions of B. Indeed, our original motivation for the study of such systems was their appearance in the study of Painleve hierarchies, where the question of the reduction of order is of great importance. However, here we do not consider such particular cases; instead we study such systems for arbitrary K, where they may represent both integrable and nonintegrable systems of ordinary differential equations. We consider the application of the Prelle-Singer (PS) method—a method used to find first integrals—to such systems in order to reduce their order. We consider the cases of coupled second order ODEs and coupled third order ODEs, as well a...