Andrew Pickering

Nonisospectral scattering problems: A key to integrable hierarchies

We show that certain partial differential equations associated to nonisospectral scattering problems in 2+1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations. This is illustrated using (an extension of) a known second-order and two new third-order nonisospectral scattering problems. These scattering problems allow us to derive new hierarchies of integrable partial differential equations, in both 1+1 and 2+1 dimensions, together with their underlying linear problems (isospectral and nonisospectral); and also new hierarchies of integrable ordinary differential equations, again with their underlying linear problems.

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlev? hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painlev? equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev? analysis to ordinary differential equations. We extend these techniques in order to derive auto-B?cklund transformations for the second Painlev? hierarchy. We also derive a number of other B?cklund transformations, including a B?cklund transformation onto a hierarchy of equations, and a little known B?cklund transformation for itself. We then use our results on B?cklund transformations to obtain, for each member of the hierarchy, a sequence of special integrals.

Rational solutions for Schwarzian integrable hierarchies

We give an approach to finding rational solutions of completely intagrable hierarchies, which makes use of the relationship between modifications and the Schwarzian equations obtained via the singular manifold method. This extends the recent work of Kudryashov, which allowed a simple derivation of the iteration used to construct sequences of such solutions. We also give a closed form for the index polynomial of the Schwarzian Korteweg-de Vries hierarchy. In addition we consider the representation of rational solutions using lower families of the hierarchy. We give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy. This representation also allows the inclusion of arbitrary data corresponding to negative indices. We use our method to derive an alternative form of the Backlund transformation for the hierarchy of the second Painleve equation, as well as new solutions of a hierarchy of breaking soliton equations. We also present here for the first time a Schwarzian version of this breaking soliton hierarchy.

Multicomponent equations associated to non-isospectral scattering problems

We consider a non-isospectral scattering problem having as its spatial part an energy-dependent Schrodinger operator. This gives rise to new completely integrable multicomponent systems of equations in (2 + 1) dimensions. Their reductions to systems in (1 + 1) dimensions have isospectral scattering problems and include multicomponent extensions of the AKNS equation and also a generalization of the Dym equation. An extension of the Fuchssteiner - Fokas - Camassa - Holm equation to (2 + 1) dimensions is also presented.

On a new non-isospectral variant of the Boussinesq hierarchy

We give a new non-isospectral extension to 2 + 1 dimensions of the Boussinesq hierarchy. Such a non-isospectral extension of the third-order scattering problem xxx +U x +(V - ) = 0 has not been considered previously. This extends our previous results on one-component hierarchies in 2 + 1 dimensions associated to third-order non-isospectral scattering problems. We characterize our entire (2 + 1)-dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non-isospectral scattering problem. This then allows an alternative approach to the construction of linear problems for the entire (2 + 1)-dimensional hierarchy. Reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems. In particular, we obtain a `fourth Painleve hierarchy, i.e. a hierarchy of ordinary differential equations having the fourth Painleveequation as its first member. We also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to Cosgrove.

Mappings preserving locations of movable poles: II. The third and fifth Painlevé equations

In a recent paper (Gordoa P R et al 1999 Nonlinearity 12 955-68) we presented a new method of deriving Backlund transformations (BTs) for ordinary differential equations. The method is based on a consideration of mappings that preserve a natural subset of movable poles, together with a careful asymptotic analysis of the transformed equation, near each type of pole. In our original paper we applied this approach to the second and fourth Painleve equations, and in a recent short paper (Gordoa P R et al 2001 Glasgow Math. J. to appear) we gave preliminary results for the third and fifth Painleve equations. Here we give full results for the latter two equations. For the third Painleve equation with γδ¬ = 0 we obtain all fundamental BTs. We also obtain a new general formulation of the second iterates of BTs that includes both γδ = 0 and γδ¬ = 0. For the fifth Painleve equation we obtain all known (non-trivial) BTs. In addition, we obtain BTs relating these two Painleve equations, as well as BTs to equations of second order and second degree, and special integrals.

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleveclassification.

Coalescence limits for higher order Painlevé equations

It is well-known that the first Painleve equation arises as a coalescence limit of each of the other five Painleve equations. This result is important because it shows that, since the solution of the first Painleve equation cannot be expressed in terms of known functions, then neither can the solutions of the other five Painleve equations (except possibly for special values of their parameters). Here we derive analogous results for three recently derived higher order ordinary differential equations believed to define new transcendental functions. We show that each of the equations considered has as a coalescence limit a member of the first Painleve hierarchy. We thus reduce the problem of showing that the solutions of these three cannot be expressed in terms of known functions to that of showing that the same is true for the corresponding first Painleve equations. This represents the first extension of coalescence results for the Painleve equations to their higher order analogues.

New integrable equations of fourth order and higher degree related to Cosgrove’s equation

We give a general formulation of the algorithm of Fokas and Ablowitz, which then allows us to obtain transformations for nth order ordinary differential equations, to equations of the same order but perhaps of higher degree. Previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by Painleve and co-workers, either to other equations in the Painleve classification or to equations of second order and second degree. As an example of our approach we consider a new fourth order ordinary differential equation due to Cosgrove which is believed to define a new transcendent. We obtain transformations relating this equation to other fourth order ordinary differential equations, of degrees ⩾2. All of these transformations, as well as the corresponding higher degree differential equations, all of which have the Painleve property, are new.

New Bäcklund transformations for the third and fourth Painlevé equations to equations of second order and higher degree

Abstract We give new Backlund transformations for the third and fourth Painleve equations, to equations of second order and higher degree.