549 research outputs found
Combined Laplace Transform-Homotopy Perturbation Method for Sine-Gordon Equation
In this paper, the combined Laplace transform-homotopy perturbation method C(LT-HPM) is presented and used to solve the initial value problem for the sine-Gordon equation to obtain the approximate-exact solutions. The results obtained show the reliability and the efficiency of this method
Instanton Moduli and Topological Soliton Dynamics
It has been proposed by Atiyah and Manton that the dynamics of Skyrmions may
be approximated by motion on a finite dimensional manifold obtained from the
moduli space of SU(2) Yang-Mills instantons. Motivated by this work we describe
how similar results exist for other soliton and instanton systems. We describe
in detail two examples for the approximation of the infinite dimensional
dynamics of sine-Gordon solitons by finite dimensional dynamics on a manifold
obtained from instanton moduli. In the first example we use the moduli space of
CP1 instantons and in the second example we use the moduli space of SU(2)
Yang-Mills instantons. The metric and potential functions on these manifolds
are constructed and the resulting dynamics is compared with the explicit exact
soliton solutions of the sine-Gordon theory.Comment: uuencoded tex file, 27 pages including 4 figures, requires phyzzx
macro. DAMTP 94-5
A symmetry breaking mechanism for selecting the speed of relativistic solitons
We propose a mechanism for fixing the velocity of relativistic soliton based
on the breaking of the Lorentz symmetry of the sine-Gordon (SG) model. The
proposal is first elaborated for a molecular chain model, as the simple
pendulum limit of a double pendulums chain. It is then generalized to a full
class of two-dimensional field theories of the sine-Gordon type. From a
phenomenological point of view, the mechanism allows one to select the speed of
a SG soliton just by tuning elastic couplings constants and kinematical
parameters. From a fundamental, field-theoretical point of view we show that
the characterizing features of relativistic SG solitons (existence of conserved
topological charges and stability) may be still preserved even if the Lorentz
symmetry is broken and a soliton of a given speed is selected.Comment: 23 pages, no figure
Lectures on the functional renormalization group method
These introductory notes are about functional renormalization group equations
and some of their applications. It is emphasised that the applicability of this
method extends well beyond critical systems, it actually provides us a general
purpose algorithm to solve strongly coupled quantum field theories. The
renormalization group equation of F. Wegner and A. Houghton is shown to resum
the loop-expansion. Another version, due to J. Polchinski, is obtained by the
method of collective coordinates and can be used for the resummation of the
perturbation series. The genuinely non-perturbative evolution equation is
obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants
of this scheme are presented where the scale which determines the order of the
successive elimination of the modes is extracted from external and internal
spaces. The renormalization of composite operators is discussed briefly as an
alternative way to arrive at the renormalization group equation. The scaling
laws and fixed points are considered from local and global points of view.
Instability induced renormalization and new scaling laws are shown to occur in
the symmetry broken phase of the scalar theory. The flattening of the effective
potential of a compact variable is demonstrated in case of the sine-Gordon
model. Finally, a manifestly gauge invariant evolution equation is given for
QED.Comment: 47 pages, 11 figures, final versio
Approximate analytical solutions of systems of PDEs by homotopy analysis method
AbstractIn this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions
Conservation laws of scaling-invariant field equations
A simple conservation law formula for field equations with a scaling symmetry
is presented. The formula uses adjoint-symmetries of the given field equation
and directly generates all local conservation laws for any conserved quantities
having non-zero scaling weight. Applications to several soliton equations,
fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein
gravitational field equations are considered.Comment: 18 pages, published version in J. Phys. A:Math. and Gen. (2003).
Added discussion of vorticity conservation laws for fluid flow; corrected
recursion formula and operator for vector mKdV conservation law
Analytical solution for cauchy reaction-diffusion problems by homotopy perturbation method
In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the Cauchy reaction-diffusion problems. HPM yields solutions in convergent series forms with easily computable terms. The HPM is tested for several examples. Comparisons of the results obtained by the HPM with that obtained by the Adomian decomposition method (ADM), homotopy analysis method (HAM) and the exact solutions show the efficiency of HPM
- …