6,552 research outputs found
Complex plane representations and stationary states in cubic and quintic resonant systems
Weakly nonlinear energy transfer between normal modes of strongly resonant
PDEs is captured by the corresponding effective resonant systems. In a previous
article, we have constructed a large class of such resonant systems (with
specific representatives related to the physics of Bose-Einstein condensates
and Anti-de Sitter spacetime) that admit special analytic solutions and an
extra conserved quantity. Here, we develop and explore a complex plane
representation for these systems modelled on the related cubic Szego and LLL
equations. To demonstrate the power of this representation, we use it to give
simple closed form expressions for families of stationary states bifurcating
from all individual modes. The conservation laws, the complex plane
representation and the stationary states admit furthermore a natural
generalization from cubic to quintic nonlinearity. We demonstrate how two
concrete quintic PDEs of mathematical physics fit into this framework, and thus
directly benefit from the analytic structures we present: the quintic nonlinear
Schroedinger equation in a one-dimensional harmonic trap, studied in relation
to Bose-Einstein condensates, and the quintic conformally invariant wave
equation on a two-sphere, which is of interest for AdS/CFT-correspondence.Comment: v2: version accepted for publicatio
Solvable cubic resonant systems
Weakly nonlinear analysis of resonant PDEs in recent literature has generated
a number of resonant systems for slow evolution of the normal mode amplitudes
that possess remarkable properties. Despite being infinite-dimensional
Hamiltonian systems with cubic nonlinearities in the equations of motion, these
resonant systems admit special analytic solutions, which furthermore display
periodic perfect energy returns to the initial configurations. Here, we
construct a very large class of resonant systems that shares these properties
that have so far been seen in specific examples emerging from a few standard
equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear
wave equations in Anti-de Sitter spacetime). Our analysis provides an
additional conserved quantity for all of these systems, which has been
previously known for the resonant system of the two-dimensional
Gross-Pitaevskii equation, but not for any other cases.Comment: v2: 23 pages, 1 figure, minor corrections, published versio
Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates
The Lowest Landau Level (LLL) equation emerges as an accurate approximation
for a class of dynamical regimes of Bose-Einstein Condensates (BEC) in
two-dimensional isotropic harmonic traps in the limit of weak interactions.
Building on recent developments in the field of spatially confined extended
Hamiltonian systems, we find a fully nonlinear solution of this equation
representing periodically modulated precession of a single vortex. Motions of
this type have been previously seen in numerical simulations and experiments at
moderately weak coupling. Our work provides the first controlled analytic
prediction for trajectories of a single vortex, suggests new targets for
experiments, and opens up the prospect of finding analytic multi-vortex
solutions.Comment: v2: minor improvements, published in PR
The Implicit Function Theorem for continuous functions
In the present paper we obtain a new homological version of the implicit
function theorem and some versions of the Darboux theorem. Such results are
proved for continuous maps on topological manifolds. As a consequence, some
versions of these classic theorems are proved when we consider differenciable
(not necessarily C^1) maps.Comment: 9 pages, no figure
Ripensando il rapporto tra il diritto della concorrenza e la contrattazione collettiva relativa al lavoro autonomo all’indomani della l. n. 81 del 2017 = Rethinking the relationship between competition law and collective bargaining relating to self-employment in the aftermath of Law n. 81 of 2017. WP C.S.D.L.E. “Massimo D’Antona”.IT – 358/2018
The essay concerns the relation between antitrust law and the collective agreements fixing the price of the services provided by self-employed workers. The Author focused on the well-known judgment of the European Court of Justice on the Dutch musicians and he contends that the antitrust exemption ought to be extended beyond the cases of employment and “false self-employment” (in its proper meaning). The argument is that, especially in the aftermath of the Italian Self-Employment Statute (Act No. 81 of 2017), the principle of freedom of union association, as enshrined at national level under article 39 of the Italian Constitution, should be construed as a fundamental social right of any individual who carries out personally or predominantly a working activity, thus shielded from any market-related evaluation
The Endorsement and Certification Committee of the European Society for Vascular Surgery (ECC-ESVS)
Covid-19 and labour law in Italy
This article provides an account of the Italian response to the Covid-19 pandemic in the labour law
field. The author focuses on the policy measures in the matters of income support, parental leave,
rest and holiday leave, agile working (i.e. teleworking), dismissal, as well as on the special provisions
arranged by the social partners and later adopted by the legislator to preserve the health and safety
of the employees and also to prevent the spread of coronavirus in the workplace. Ultimately, the
author hints at the potential development of employee participation in Italy in the wake of the
upsurge of social dialogue during the coronavirus emergency
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