94 research outputs found
Probing analytical and numerical integrability: The curious case of
Motivated by recent studies related to integrability of string motion in
various backgrounds via analytical and numerical procedures, we discuss these
procedures for a well known integrable string background . We start by revisiting conclusions from earlier studies on string
motion in and and then move on
to more complex problems of and
. Discussing both analytically and numerically, we deduce that
while strings do not encounter any irregular trajectories,
string motion in the deformed five-sphere can indeed, quite surprisingly, run
into chaotic trajectories. We discuss the implications of these results both on
the procedures used and the background itself.Comment: 31 pages, 3 figures, references updated, analysis for Spiky strings
in section (4.1) have been revised, version to appear in JHE
On Marginal Deformations and Non-Integrability
We study the interplay between a particular marginal deformation of super Yang-Mills theory, the deformation, and integrability in
the holographic setting. Using modern methods of analytic non-integrability of
Hamiltonian systems, we find that, when the parameter takes imaginary
values, classical string trajectories on the dual background become
non-integrable. We expect the same to be true for generic complex
parameter. By exhibiting the Poincar\'e sections and phase space trajectories
for the generic complex case, we provide numerical evidence of strong
sensitivity to initial conditions. Our findings agree with expectations from
weak coupling that the complex deformation is non-integrable and
provide a rigorous argument beyond the trial and error approach to
non-integrability.Comment: 19 pages, 9 figure
Non-integrability in non-relativistic theories
Generic non-relativistic theories giving rise to non-integrable string
solutions are classified. Our analysis boils down to a simple algebraic
condition for the scaling parameters of the metric. Particular cases are the
Lifshitz and the anisotropic Lifshitz spacetimes, for which we find that for
trivial dilaton dependence the only integrable physical theory is that for z=1.
For the hyperscaling violation theories we conclude that the vast majority of
theories are non-integrable, while only for a small class of physical theories,
where the Fermi surfaces belong to, integrability is not excluded. Schrodinger
theories are also analyzed and a necessary condition for non-integrability is
found. Our analysis is also applied to cases where the exponential of the
dilaton is a monomial of the holographic coordinate.Comment: 1+20 pages, v2:minor corrections, references adde
Finding nonlocal Lie symmetries algorithmically
Here we present a new approach to compute symmetries of rational second order
ordinary differential equations (rational 2ODEs). This method can compute Lie
symmetries (point symmetries, dynamical symmetries and non-local symmetries)
algorithmically. The procedure is based on an idea arising from the formal
equivalence between the total derivative operator and the vector field
associated with the 2ODE over its solutions (Cartan vector field). Basically,
from the formal representation of a Lie symmetry it is possible to extract
information that allows to use this symmetry practically (in the 2ODE
integration process) even in cases where the formal operation cannot be
performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the
2ODE in question depends on parameters, the procedure allows an analysis that
determines the regions of the parameter space in which the integrable cases are
located
Strongly formal Weierstrass non-integrability for polynomial differential systems in C2
Recently a criterion has been given for determining the weakly formal
Weierstrass non-integrability of polynomial differential systems in C2
. Here we extend this criterion for determining the strongly formal Weierstrass non-integrability
which includes the weakly formal Weierstrass non-integrability of polynomial differential systems in C2
. The criterion is based on the solutions of the form y = f(x) with
f(x) ∈ C[[x]] of the differential system whose integrability we are studying. The results
are applied to a differential system that contains the famous force-free Duffing and the
Duffing–Van der Pol oscillators.The first author is partially supported by a MINECO/ FEDER grant number MTM2017-84383- P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR-1276. The second author is partially supported by the Ministerio de EconomÃa, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER) and grant MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911
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