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

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