A. Ramani

The Painlevé property and singularity analysis of integrable and non-integrable systems

Abstract We present a review of results of the so-called Painleve singularity approach to the investigation of the integrability of dynamical systems with finite and infinite number of degrees of freedom. Rigorous results based on the theorems of Yoshida and Ziglin concerning proofs of non-integrability are also presented, as well as an application of the new “poly-Painleve” method due to Kruskal. Finally a section is devoted to the singularity analysis of the solutions of non-integrable dynamical systems.

Structural stability of the Korteweg-De Vries solitons under a singular perturbation

Abstract We investigate the stability of a solitary wave solution of the Korteweg-de Vries equation when a fifth order spatial derivative term is added. We show that the solution ceases to be strictly localized but develops an infinite oscillating tail and we compute the amplitude of the latter.

Discrete Painlevé equations: coalescences, limits and degeneracies

Starting from the standard form of the five discrete Painleve equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painleve equations. A particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n-dependence and introduces more new discrete Painleve equations.

Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable?

The Kadomtsev–Petviashvili (KP) hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy. Since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the KP hierarchy provides integrable scalar equations with an arbitrary number of independent variables. It is shown, by investigating a specific equation in 3+1 dimensions, that the higher equations in the KP hierarchy are only integrable in a conditional sense. The equation under study, taken in isolation, does not pass certain well‐known and reliable integrability tests. Thus, applying Painleve analysis, we find that solutions exist, allowing movable critical points. Furthermore, solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions. On the other hand, if the dependence of a solution on the first 2+1 variables is restricted by the fact t...

A new class of integrable systems

We present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation (PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak‐Painleve property’’).

Integrals of quadratic ordinary differential equations in R3: The Lotka-Volterra system

A method already introduced by the last two authors for finding the integrable cases of three-dimensional autonomous ordinary differential equations based on the Frobenius integrability theorem is described in detail. Using this method and computer algebra, the so-called three-dimensional Lotka-Volterra system is studied. Many cases of integrability are thus found. The study of this system is completed by the application of Painleve analysis and the Jacobi last multiplier method. The methods used are of general interest and can be applied to many other systems.

On the complete and partial integrability of non-Hamiltonian systems

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painleve property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painleve transcendents. Relaxing the Painleve property we find many partially integrable cases whose movable singularities are poles at leading order, with In(t-t0) terms entering at higher orders. In an Nth order, generalized Rossler model a precise relation is established between the partial fulfillment of the Painleve conditions and the existence of N - 2 integrals of the motion.

Integrability of Hamiltonians with third‐ and fourth‐degree polynomial potentials

The weak‐Painleve property, as a criterion of integrability, is applied to the case of simple Hamiltonians describing the motion of a particle in two‐dimensional polynomial potentials of degree three and four. This allows a complete identification of all the integrable cases of cubic potentials. In the case of quartic potentials, although our results are not exhaustive, some new integrable cases are discovered. In both cases the integrability is explicited by a direct calculation of the second integral of motion of the system.

Isomonodromic deformation problems for discrete analogues of Painlevé equations

Abstract Starting from isospectral problems of two-dimensional integrable mappings, isomonodromic deformation problems for the new descrete versions of the Painleve I–III equations are constructed. It is argued that they lead to differential and q-difference deformations (“quantizations”) of the corresponding spectral curves.

Integrable lattices and convergence acceleration algorithms

Abstract We show that a well-known convergence acceleration scheme, the ϵ-algorithm, when viewed as a two-variable difference equation, is nothing but the discrete Korteweg-de Vries lattice equation. The complete integrability of the latter confers to the ϵ-algorithm interesting properties among which the singularity confinement is outstanding. In fact, this property is used in order to derive the generalizations of the ϵ-accelerator leading to the most general form of the ϱ-algorithm. A new acceleration algorithm based on the modified Korteweg-de Vries lattice equation is also derived.