4,212 research outputs found
Weak Lax pairs for lattice equations
We consider various 2D lattice equations and their integrability, from the
point of view of 3D consistency, Lax pairs and B\"acklund transformations. We
show that these concepts, which are associated with integrability, are not
strictly equivalent. In the course of our analysis, we introduce a number of
black and white lattice models, as well as variants of the functional
Yang-Baxter equation
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
Semiclassical approach to Bose-Einstein condensates in a triple well potential
We present a new approach for the analysis of Bose-Einstein condensates in a
few mode approximation. This method has already been used to successfully
analyze the vibrational modes in various molecular systems and offers a new
perspective on the dynamics in many particle bosonic systems. We discuss a
system consisting of a Bose-Einstein condensate in a triple well potential.
Such systems correspond to classical Hamiltonian systems with three degrees of
freedom. The semiclassical approach allows a simple visualization of the
eigenstates of the quantum system referring to the underlying classical
dynamics. From this classification we can read off the dynamical properties of
the eigenstates such as particle exchange between the wells and entanglement
without further calculations. In addition, this approach offers new insights
into the validity of the mean-field description of the many particle system by
the Gross-Pitaevskii equation, since we make use of exactly this correspondence
in our semiclassical analysis. We choose a three mode system in order to
visualize it easily and, moreover, to have a sufficiently interesting
structure, although the method can also be extended to higher dimensional
systems.Comment: 15 pages, 15 figure
On the relevance of chaos for halo stars in the Solar Neighbourhood
We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun
may not be as relevant as previously suggested in erasing phase space
signatures of past Galactic accretion events. For this purpose, we analyse
Solar Neighbourhood-like volumes extracted from cosmological simulations that
naturally account for chaotic orbital behaviour induced by the strongly
triaxial and cuspy shape of the resulting dark matter haloes, among other
factors. In the approximation of an analytical static triaxial model, our
results show that a large fraction of stellar halo particles in such local
volumes have chaos onset times (i.e., the timescale at which stars commonly
associated with chaotic orbits will exhibit their chaotic behaviour)
significantly larger than a Hubble time. Furthermore, particles that do present
a chaotic behaviour within a Hubble time do not exhibit significant diffusion
in phase space.Comment: 20 pages, 16 figures. Accepted for publication in MNRA
Kink dynamics in the MSTB model
ProducciĂłn CientĂficaIn this paper kink scattering processes are investigated in the Montonen–Sarker–Trullinger–Bishop (MSTB) model. The MSTB model is in fact a one-parametric family of relativistic scalar field theories living in a one-time one-space Minkowski space-time which encompasses two coupled scalar fields. Among the static solutions of the model two kinds of topological kinks are distinguished in a precise range of the family parameter. In that regime there exists one unstable kink exhibiting only one non-null component of the scalar field. Another type of topological kink solutions, stable in this case, includes two different kinks for which the two components of the scalar field are non-null. Both one-component and two-component topological kinks are accompanied by their antikink partners. The decay of the unstable kink to one of the stable solutions plus radiation is numerically computed. The pair of stable two-component kinks living
respectively on upper and lower semi-ellipses in the field space belongs to the same topological sector in the configuration space and provides an ideal playground to address several scattering events involving one kink and either its own antikink or the antikink of the other stable kink. By means of numerical analysis we shall find and describe interesting physical phenomena. Bion (kink–antikink oscillations) formation, kink reflection, kink–antikink annihilation, kink transmutation and resonances are examples of these types of events. The appearance of these phenomena emerging in the kink–antikink scattering depends critically on the initial collision velocity and the chosen value of the coupling constant parametrizing the family of MSTB models.MINDECO grant MTM2014-57129-C2-1-P and Junta de Castilla y León grants VA057U16 and BU229P18
Special Lagrangian m-folds in C^m with symmetries
This is the first in a series of papers on special Lagrangian submanifolds in
C^m. We study special Lagrangian submanifolds in C^m with large symmetry
groups, and give a number of explicit constructions. Our main results concern
special Lagrangian cones in C^m invariant under a subgroup G in SU(m)
isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an
o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct,
G-invariant special Lagrangian cones on T^{m-1} in C^m.
These examples are interesting as local models for singularities of special
Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to
understand Mirror Symmetry and the SYZ conjecture.Comment: 44 pages, LaTeX; (v4) minor corrections and improvement
Adiabatically coupled systems and fractional monodromy
We present a 1-parameter family of systems with fractional monodromy and
adiabatic separation of motion. We relate the presence of monodromy to a
redistribution of states both in the quantum and semi-quantum spectrum. We show
how the fractional monodromy arises from the non diagonal action of the
dynamical symmetry of the system and manifests itself as a generic property of
an important subclass of adiabatically coupled systems
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