3,804 research outputs found
Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation
The main objective for this research was the analytical exploration of the
dynamics of planar satellite rotation during the motion of an elliptical orbit
around a planet. First, we revisit the results of J. Wisdom et al. (1984), in
which, by the elegant change of variables (considering the true anomaly f as
the independent variable), the governing equation of satellite rotation takes
the form of an Abel ODE of the second kind, a sort of generalization of the
Riccati ODE. We note that due to the special character of solutions of a
Riccati-type ODE, there exists the possibility of sudden jumping in the
magnitude of the solution at some moment of time. In the physical sense, this
jumping of the Riccati-type solutions of the governing ODE could be associated
with the effect of sudden acceleration/deceleration in the satellite rotation
around the chosen principle axis at a definite moment of parametric time. This
means that there exists not only a chaotic satellite rotation regime (as per
the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe
(Arnold 1992) could occur during the satellite rotation process. We especially
note that if a gradient catastrophe could occur, this does not mean that it
must occur: such a possibility depends on the initial conditions. In addition,
we obtained asymptotical solutions that manifest a quasi-periodic character
even with the strong simplifying assumptions e ~ 0, p = 1, which reduce the
governing equation of J. Wisdom et al. (1984) to a kind of Beletskii equation.Comment: 15 pages; Keywords: Beletskii equation, satellite rotation, Abel ODE,
gradient catastrophe. Article is published in Theoretical and Applied
Mechanics Letters (7 June 2017
Time-like boundary conditions in the NLS model
We focus on the non-linear Schrodinger model and we extend the notion of
space-time dualities in the presence of integrable time-like boundary
conditions. We identify the associated time-like `conserved' quantities and Lax
pairs as well as the corresponding boundary conditions. In particular, we
derive the generating function of the space components of the Lax pairs in the
case of time-like boundaries defined by solutions of the reflection equation.
Analytical conditions on the boundary Lax pair lead to the time like-boundary
conditions. The time-like dressing is also performed for the first time, as an
effective means to produce the space components of the Lax pair of the
associated hierarchy. This is particularly relevant in the absence of a
classical r-matrix, or when considering complicated underlying algebraic
structures. The associated time Riccati equations and hence the time-like
conserved quantities are also derived. We use as the main paradigm for this
purpose the matrix NLS-type hierarchy.Comment: 17 pages, LaTex. A few typos corrected. arXiv admin note: substantial
text overlap with arXiv:1810.1093
A Periodic Systems Toolbox for MATLAB
The recently developed Periodic Systems Toolbox for MATLAB is described. The basic approach to develop this toolbox was to exploit the powerful object manipulation features of MATLAB via flexible andfunctionally rich high level m-functions, while simultaneously enforcing highly efficient and numerically sound computations via the mex-function technology of MATLAB to solve critical numerical problems.The m-functions based user interfaces ensure user-friendliness in operating with the functions of this toolbox via an object oriented approach to handle periodic system descriptions. The mex-functions are based on Fortran implementations of recently developed structure exploiting and structure preserving numerical algorithms for periodic systems which completely avoid forming of lifted representations
Integrals of Motion in the Two Killing Vector Reduction of General Relativity
We apply the inverse scattering method to the midi-superspace models that are
characterized by a two-parameter Abelian group of motions with two spacelike
Killing vectors. We present a formulation that simplifies the construction of
the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us
to obtain the zero curvature formulation for these models. Using this, and
imposing periodic boundary conditions corresponding to the Gowdy models when
the spatial topology is a three torus , we show that the equation of
motion for the monodromy matrix is an evolution equation of the Heisenberg
type. Consequently, the eigenvalues of the monodromy matrix are the generating
functionals for the integrals of motion. Furthermore, we utilise a suitable
formulation of the transition matrix to obtain explicit expressions for the
integrals of motion. This involves recursion relations which arise in solving
an equation of Riccati type. In the case when the two Killing vectors are
hypersurface orthogonal the integrals of motion have a particularly simple
form.Comment: 20 pages, plain TeX, SU-GP-93/7-8, UM-P-93/7
Hydrodynamic mean field solutions of 1D exclusion processes with spatially varying hopping rates
We analyze the open boundary partially asymmetric exclusion process with
smoothly varying internal hopping rates in the infinite-size, mean field limit.
The mean field equations for particle densities are written in terms of Ricatti
equations with the steady-state current as a parameter. These equations are
solved both analytically and numerically. Upon imposing the boundary conditions
set by the injection and extraction rates, the currents are found
self-consistently. We find a number of cases where analytic solutions can be
found exactly or approximated. Results for from asymptotic analyses for
slowly varying hopping rates agree extremely well with those from extensive
Monte Carlo simulations, suggesting that mean field currents asymptotically
approach the exact currents in the hydrodynamic limit, as the hopping rates
vary slowly over the lattice. If the forward hopping rate is greater than or
less than the backward hopping rate throughout the entire chain, the three
standard steady-state phases are preserved. Our analysis reveals the
sensitivity of the current to the relative phase between the forward and
backward hopping rate functions.Comment: 12 pages, 4 figure
Linking Backlund and Monodromy Charges for Strings on AdS_5 x S^5
We find an explicit relation between the two known ways of generating an
infinite set of local conserved charges for the string sigma model on AdS_5 x
S^5: the Backlund and monodromy approaches. We start by constructing the
two-parameter family of Backlund transformations for the string with an
arbitrary world-sheet metric. We then show that only for a special value of one
of the parameters the solutions generated by this transformation are compatible
with the Virasoro constraints. By solving the Backlund equations in a
non-perturbative fashion, we finally show that the generating functional of the
Backlund conservation laws is equal to a certain sum of the quasi-momenta. The
positions of the quasi-momenta in the complex spectral plane are uniquely
determined by the real parameter of the Backlund transform.Comment: 25 pages, 1 figur
Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
The existence of a Lagrangian description for the second-order Riccati
equation is analyzed and the results are applied to the study of two different
nonlinear systems both related with the generalized Riccati equation. The
Lagrangians are nonnatural and the forces are not derivable from a potential.
The constant value of a preserved energy function can be used as an
appropriate parameter for characterizing the behaviour of the solutions of
these two systems. In the second part the existence of two--dimensional
versions endowed with superintegrability is proved. The explicit expressions of
the additional integrals are obtained in both cases. Finally it is proved that
the orbits of the second system, that represents a nonlinear oscillator, can be
considered as nonlinear Lissajous figuresComment: 25 pages, 7 figure
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