66 research outputs found
Jacobi-Maupertuis metric of Lienard type equations and Jacobi Last Multiplier
We present a construction of the Jacobi-Maupertuis (JM) principle for an
equation of the Lienard type, viz \ddot{x} + f(x)x^2 + g(x) = 0 using Jacobi's
last multiplier. The JM metric allows us to reformulate the Newtonian equation
of motion for a variable mass as a geodesic equation for a Riemannian metric.
We illustrate the procedure with examples of Painleve-Gambier XXI, the Jacobi
equation and the Henon-Heiles system
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches
In this paper we employ three recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a family of
so-called short-pulse equations (SPE). A recent, novel application of
phase-plane analysis is first employed to show the existence of breaking kink
wave solutions in certain parameter regimes. Secondly, smooth traveling waves
are derived using a recent technique to derive convergent multi-infinite series
solutions for the homoclinic (heteroclinic) orbits of the traveling-wave
equations for the SPE equation, as well as for its generalized version with
arbitrary coefficients. These correspond to pulse (kink or shock) solutions
respectively of the original PDEs.
Unlike the majority of unaccelerated convergent series, high accuracy is
attained with relatively few terms. And finally, variational methods are
employed to generate families of both regular and embedded solitary wave
solutions for the SPE PDE. The technique for obtaining the embedded solitons
incorporates several recent generalizations of the usual variational technique
and it is thus topical in itself. One unusual feature of the solitary waves
derived here is that we are able to obtain them in analytical form (within the
assumed ansatz for the trial functions). Thus, a direct error analysis is
performed, showing the accuracy of the resulting solitary waves. Given the
importance of solitary wave solutions in wave dynamics and information
propagation in nonlinear PDEs, as well as the fact that not much is known about
solutions of the family of generalized SPE equations considered here, the
results obtained are both new and timely.Comment: accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
Separation of variables for a lattice integrable system and the inverse problem
We investigate the relation between the local variables of a discrete
integrable lattice system and the corresponding separation variables, derived
from the associated spectral curve. In particular, we have shown how the
inverse transformation from the separation variables to the discrete lattice
variables may be factorised as a sequence of canonical transformations,
following the procedure outlined by Kuznetsov.Comment: 14 pages. submitted for publicatio
Dynamical Studies of Equations from the Gambier Family
We consider the hierarchy of higher-order Riccati equations and establish their connection with the Gambier equation. Moreover we investigate the relation of equations of the Gambier family to other nonlinear differential systems. In particular we explore their connection to the generalized Ermakov-Pinney and Milne-Pinney equations. In addition we investigate the consequence of introducing Okamoto's folding transformation which maps the reduced Gambier equation to a Liénard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored
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