94,129 research outputs found
Formal analytical solutions for the Gross-Pitaevskii equation
Considering the Gross-Pitaevskii integral equation we are able to formally
obtain an analytical solution for the order parameter and for the
chemical potential as a function of a unique dimensionless non-linear
parameter . We report solutions for different range of values for the
repulsive and the attractive non-linear interactions in the condensate. Also,
we study a bright soliton-like variational solution for the order parameter for
positive and negative values of . Introducing an accumulated error
function we have performed a quantitative analysis with other well-established
methods as: the perturbation theory, the Thomas-Fermi approximation, and the
numerical solution. This study gives a very useful result establishing the
universal range of the -values where each solution can be easily
implemented. In particular we showed that for , the bright soliton
function reproduces the exact solution of GPE wave function.Comment: 8 figure
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Analytical Approximation Methods for the Stabilizing Solution of the HamiltonâJacobi Equation
In this paper, two methods for approximating the stabilizing solution of the HamiltonâJacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the HamiltonâJacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
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