113 research outputs found
General solution for Hamiltonians with extended cubic and quartic potentials
We integrate with hyperelliptic functions a two-particle Hamiltonian with
quartic potential and additionnal linear and nonpolynomial terms in the
Liouville integrable cases 1:6:1 and 1:6:8.Comment: LaTex 2e. To appear, Theoretical and Mathematical Physics 200
On the exact solutions of the Bianchi IX cosmological model in the proper time
It has recently been argued that there might exist a four-parameter analytic
solution to the Bianchi IX cosmological model, which would extend the
three-parameter solution of Belinskii et al. to one more arbitrary constant. We
perform the perturbative Painlev\'e test in the proper time variable, and
confirm the possible existence of such an extension.Comment: 8 pages, no figure, standard Latex, to appear in Regular and chaotic
dynamics (1998
Solitons from a direct point of view: padeons
AbstractA systematic approach to soliton interaction is presented in terms of a particular class of solitary waves (padeons) which are linear fractions with respect to the nonlinearity parameter ϵ. A straightforward generalization of the padeon to higher order rational fractions (multipadeon) yields a natural ansatz for N-soliton solutions. This ansatz produces multisoliton formulas in terms of an ‘interaction matrix’ A. The structure of the matrix gives some insight into the hidden IST-properties of a familiar set of ‘integrable’ equations (KdV, Boussinesq, MKdV, sine-Gordon, nonlinear Schrödinger). The analysis suggests a ‘padeon’ working definition of the soliton, leading to an explicit set of necessary conditions on the padeon equation
Integration of a generalized H\'enon-Heiles Hamiltonian
The generalized H\'enon-Heiles Hamiltonian
with an additional
nonpolynomial term is known to be Liouville integrable for three
sets of values of . It has been previously integrated by genus
two theta functions only in one of these cases. Defining the separating
variables of the Hamilton-Jacobi equations, we succeed here, in the two other
cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic
A q-analogue of gl_3 hierarchy and q-Painleve VI
A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a
reduction of the q-KP hierarchy. Applying a similarity reduction and a
q-Laplace transformation to the hierarchy, one can obtain the q-Painleve VI
equation proposed by Jimbo and Sakai.Comment: 14 pages, IOP style, to appear in J. Phys. A Special issue "One
hundred years of Painleve VI
The Bianchi Ix (MIXMASTER) Cosmological Model is Not Integrable
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 Painlev\'e test; this proves the inexistence of any
vacuum solution other than the three known ones.Comment: 14 pages, no figure
Solitary waves of nonlinear nonintegrable equations
Our goal is to find closed form analytic expressions for the solitary waves
of nonlinear nonintegrable partial differential equations. The suitable
methods, which can only be nonperturbative, are classified in two classes.
In the first class, which includes the well known so-called truncation
methods, one \textit{a priori} assumes a given class of expressions
(polynomials, etc) for the unknown solution; the involved work can easily be
done by hand but all solutions outside the given class are surely missed.
In the second class, instead of searching an expression for the solution, one
builds an intermediate, equivalent information, namely the \textit{first order}
autonomous ODE satisfied by the solitary wave; in principle, no solution can be
missed, but the involved work requires computer algebra.
We present the application to the cubic and quintic complex one-dimensional
Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.Comment: 28 pages, chapter in book "Dissipative solitons", ed. Akhmediev, to
appea
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