John Weiss

The Painlevé property for partial differential equations

In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

THE PAINLEVE PROPERTY FOR PARTIAL DIFFERENTIAL EQUATIONS. II. BACKLUND TRANSFORMATION, LAX PAIRS, AND THE SCHWARZIAN DERIVATIVE

In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.

On classes of integrable systems and the Painlevé property

The Caudrey–Dodd–Gibbon equation is found to possess the Painleve property. Investigation of the Backlund transformations for this equation obtains the Kuperschmidt equation. A certain transformation between the Kuperschmidt and Caudrey–Dodd–Gibbon equation is obtained. This transformation is employed to define a class of p.d.e.’s that identically possesses the Painleve property. For equations within this class Backlund transformations and rational solutions are investigated. In particular, the sequences of higher order KdV, Caudrey–Dobb–Gibbon, and Kuperschmidt equations are shown to possess the Painleve property.

The sine‐Gordon equations: Complete and partial integrability

The sine–Gordon equation in one space‐one time dimension is known to possess the Painleve property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Backlund transform and the Lax pair for this equation. A connection with the sequence of higher‐order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painleve. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painleve property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painleve property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space ...

The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations

We investigate the sequence of Boussinesq equations by the method of singular manifolds. For the Boussinesq equation, which is known to possess the Painleve property, a Backlund transformation is defined. This Backlund transformation, which is formulated in terms of the Schwarzian derivative, obtains the system of modified Boussinesq equations and the resulting Miura‐type transformation. The modified Boussinesq equations are found to be invariant under a discrete group of symmetries, acting on the dependent variables. By linearizing the Miura transformation (and modified equations) the Lax pair is readily obtained. Furthermore, by a result of Fokas and Anderson, the recursion operators defining the sequence of (higher‐order) Boussinesq equations may be constructed from the Miura transformation. This allows the (recursive) definition of Backlund transformations for this sequence of equations. The recursion operator is found to preserve the discrete symmetries of the modified Boussinesq equations. This lead...

Bäcklund transformation and the Painlevé property

When a differential equation possesses the Painleve property it is possible (for specific equations) to define a Backlund transformation (by truncating an expansion about the ‘‘singular’’ manifold at the constant level term). From the Backlund transformation, it is then possible to derive the Lax pair, modified equations and Miura transformations associated with the ‘‘completely integrable’’ system under consideration. In this paper completely integrable systems are considered for which Backlund transformations (as defined above) may not be directly defined. These systems are of two classes. The first class consists of equations of Toda lattice type (e.g., sine–Gordon, Bullough–Dodd equations). We find that these equations can be realized as the ‘‘minus‐one’’ equation of sequences of integrable systems. Although the ‘‘Backlund transformation’’ may or may not exist for the ‘‘minus‐one’’ equation, it is shown, for specific sequences, that the Backlund transformation does exist for the ‘‘positive’’ equations...

Periodic fixed points of Bäcklund transformations and the Korteweg–de Vries equation

A new method for studying integrable systems based on the ‘‘periodic fixed points’’ of Backlund transformations (BT’s) is presented. Normally the BT maps an ‘‘old’’ solution into a ‘‘new’’ solution and requires a known ‘‘seed’’ solution to get started. Besides this limitation, it can also be difficult to qualitatively classify the result of applying the BT several times to a known solution. By studying the periodic fixed points of the BT (regarded as a nonlinear map in a function space), integrable systems of equations of finite degree (equal to the order of the fixed point) and a method for the systematic classification of the solutions of the original system are obtained.

Modified equations, rational solutions, and the Painleve property for the Kadomtsev--Petviashvili and Hirota--Satsuma equations

We propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painleve property, a Backlund transformation is defined in terms of an expansion about the singular manifold. This Backlund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati‐type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Backlund transformations of the modified equations provides a method for the iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev–Petviashvili equation and the Hirota–Satsuma equations.

Periodic fixed points of Bäcklund transformations

The periodic fixed points of the Backlund transformations are finite dimensional invariant manifolds for the flow of the system. The dynamics occur as commuting hamiltonian flows on this finite dimensional manifold. We examine the flow of the KdV periodic fixed points in the neighborhood of steady states and reductions. These are analogous to a flow in the neighborhood of a sequence of heteroclinic points.

Bäcklund transformation and linearizations of the Henon-Heiles system

Abstract Whem the Henon-Heiles system possesses the Painleve property certain Backlund transformations are defined in terms of the manifold of singularities. The resulting system of equations are shown to effectively linearize the Henon-Heiles system.