Micheline Musette

Link between solitary waves and projective Riccati equations

Many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, the authors deduce a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. They handle several examples. For the Henon-Heiles Hamiltonian system, they obtain several exact solutions; one of them defines a new solitary wave solution for a coupled system of Boussinesq and nonlinear Schrodinger equations. For a third order dispersive equation with two monomial nonlinearities, they isolate all cases where the general solution is single valued.

Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation

Abstract Systems of two linear partial differential equations (PDEs) with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear PDEs. For the one-dimensional complex Ginzburg- Landau equation, the usual representations of the complex field (Re A , or ( A , Ā ) are multivalued, which makes difficult the search for exact solutions in this manner. Fortunately, the Painleve analysis naturally introduces an elementary single valued-like representation of A by two fields ( Z ,grad Θ), respectively complex abd real, uniquely defined by an explicit expression for Θ and A = Ze iΘ . Another important feature is the invariance by parity on A , which increases the class of expected solutions. This allows to retrieve quite easily the four famous solutions of Nozaki and Bekki, represented by the constant coefficients of two linear partial differential equations and a finite set of constants.

Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations

Given a partial differential equation, its Painleve analysis will first be performed with a built‐in invariance under the homographic group acting on the singular manifold function. Then, assuming an order for the underlying Lax pair, a multicomponent pseudopotential of projective Riccati type, the components of which are homographically invariant, is introduced. If the equation admits a classical Darboux transformation, a very small set of determining equations whose solution yields the Lax pair will be generated in the basis of the pseudopotential. This new method will be applied to find the yet unpublished Lax pair of the scalar Hirota–Satsuma equation.

The Bianchi IX (mixmaster) cosmological model is not integrable

Abstract The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative PainlevtE test; this proves the inexistence of any vacuum solution other than the three known ones.

Analytic solitary waves of nonintegrable equations

Abstract A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ordinary differential equation (ODE) satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg–Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

Painlevé Analysis for Nonlinear Partial Differential Equations

The Painleve analysis introduced by Weiss, Tabor, and, Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDEs) is an extension of the method initiated by Painleve and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODEs) without movable critical points. In this chapter we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated Backlund transformations. Many remarkable properties are shared by these so-called integrable equations, but they are generically no longer valid for equations modeling physical phenomena. Belonging to this second class, some equations called “partially integrable” sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed-form analytic solutions, which necessarily agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations that are linearizable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.

Explicit integration of the Hénon-Heiles Hamiltonians

Abstract We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painlevé sense (impossibility to add any term to the Hamiltonian without destroying the Painlevé property).

The two-singular manifold method: II. Classical Boussinesq system

For pt.I see Musette et. al., ibid., vol.24, p.3895 (1994) The two-singular manifold method-a generalization of the singular manifold method of Weiss (1983)-is applied to the classical Boussinesq system, also known as the Broer-Kaup system. From the point of view of its singularity analysis, the important feature of this system is the existence of two principal families with opposite principal parts. The usual singular manifold method takes into account only one of these families at a time. Our generalization takes into account both families, and in this way we are able to derive the Lax pair and Darboux transformation-and hence the auto-Backlund transformation-for the classical Boussinesq system from its Painleve analysis.

A new method to test discrete Painlevé equations

Abstract Necessary discretization rules to preserve the Painleve property are stated. A new method is added to the discrete Painleve test, which perturbs the continuum limit and generates infinitely many no-log conditions.

Nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation

Abstract The fifth order Kaup–Kupershmidt (KK) equation is one of the solitonic equations related to the integrable cases of the Henon–Heiles system. As opposed to the Sawada–Kotera equation which is its dual equation, the construction of the N -soliton solutions of KK is not an easy task in using a perturbation scheme or the Hirota bilinear formalism. From the Backlund transformation obtained by singularity analysis considerations, we here establish the permutability theorem for the KK equation. This allows us to explain the “anomalous” structure of the N -soliton solution, empirically obtained by Parker [Physica D 137 (2000) 25–33, 34–48].