Pilar R. Gordoa

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.

DARBOUX TRANSFORMATIONS VIA PAINLEVE ANALYSIS

The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota -function method. In this paper, we report how to improve the singular manifold method when the equation has more than one Painlev? branch. The singular manifold method generalized in such a way is applied to a pair of equations in 2 + 1 dimensions

Painleve analysis of the generalized Burgers-Huxley equation

A complete Painleve test (1900) is applied to the generalized Burgers-Huxley equation using the version of the Painleve analysis recently developed by Weiss, Tabor and Carnevale (1983) for partial nonlinear differential equations. In so doing, the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published.

Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems

We present a modified treatment of the singular manifold method as an improved variant of the Painleve analysis for partial differential equations with two branches in the Painleve expansion. We illustrate the method by fully applying it to the Classical Boussinesq system and to the Mikhailov-Shabat system.

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.

Unified approach to Miura, Backlund and Darboux Transformations for Nonlinear Partial Differential Equations

This paper is an attempt to present and discuss at some length the Singular Mani- fold Method. This Method is based upon the Painleve Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Backlund or Darboux Transformations as well as τ -functions, in a unified way. Besides to present the ba- sics of the Method we exemplify this approach by applying it to four equations in (1 + 1)-dimensions. Two of them are related with the other two through Miura trans- formations that are also derived by using the Singular Manifold Method.

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.

Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations

We recently introduced two new sixth-order partial differential equations (PDEs) associated with third-order scattering problems. Here we extend our study of these PDEs by considering the construction of exact solutions both by using the method of symmetry reduction due to Lie, and by using their Darboux transformations (DTs). Amongst the ordinary differential equations (ODEs) obtained by symmetry reduction is an ODE due to Cosgrove that is believed to define a new Painleve transcendent. This ODE provides soliton solutions for our integrable PDEs that include arbitrary functions of time. The DTs for our PDEs allow the recovery of these solutions and in addition provide other solutions which are not associated with Lie symmetries (either classical or nonclassical). We also consider the iteration of the corresponding Backlund transformations (BTs) for these PDEs. The theorem of permutability allows us to reduce this process of iterating the DT from one of solving a third-order linear equation (the spatial p...

Bäcklund transformations for the second member of the first Painlevé hierarchy

Abstract In a recent paper (J. Math. Phys. 42 (2001) 1697) we gave a general formulation of an algorithm introduced by Fokas and Ablowitz that allows us to obtain Backlund transformations for n th-order ordinary differential equations, to equations of the same order but perhaps of higher degree. Here we apply this procedure to the fourth-order ordinary differential equation which is the second member of the first Painleve hierarchy. This allows us to obtain new Backlund transformations relating this equation to other fourth-order ordinary differential equations of degrees ⩾2.