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Un nouvel algorithme pour la recherche des coordonnees de separation est teste sur l'exemple de la toupie de Kowalevski.

We study the separability of the Neumann-Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system., LaTeX file, 14 pages

We consider a quantum system of two particles in dimension three interacting via a smooth potential. We characterize the asymptotic dynamics in the limit of small mass ratio for an initial state given in product form, with an explicit control of the error. An application to the decoherence effect produced on the heavy particle is also discussed.

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.

We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating, by numerical computations, the trend towards equipartition in the thermodynamic limit. We concentrate our attention on a particular class of initial conditions, namely, with all the energy on the first mode or the first few modes. We observe that the approach to equipartition occurs on two different time scales: in a short time the energy spreads up by forming a packet involving all low--frequency modes up to a cutoff frequency $\omega_c$, while a much longer time is required in order to reach equipartition, if any. In this sense one has an energy localization with respect to frequency. The crucial point is that our numerical computations suggest that this phenomenon of a fast formation of a natural packet survives in the thermodynamic limit. More precisely we conjecture that the cutoff frequency $\omega_c$ is a function of the specific energy $\epsilon = E/N$, where $E$ and $N$ are the total energy and the number of particles, respectively. Equivalently, there should exist a function $\epsilon_c(\omega)$, representing the minimal specific energy at which the natural packet extends up to frequency $\omega$. The time required for the fast formation of the natural packet is also investigated.

A simple example shows the unexpected role of Lenard chains in the theory of classical integrable systems.

Geometric and topological aspects associated with induction effects of field lines in the shape of torus knots/unknots are examined and discussed in detail. Knots are assumed to lie on a mathematical torus of circular cross-section and are parametrized by standard equations. The induced field is computed by direct integration of the Biot-Savart law. Field line patterns of the induced field are obtained and several properties are examined for a large family of knots/unknots up to 51 crossings. The intensity of the induced field at the origin of the reference system (center of the torus) is found to depend linearly on the number of toroidal coils and reaches maximum values near the boundary of the mathematical torus. New analytical estimates and bounds on energy and helicity are established in terms of winding number and minimum crossing number. These results find useful applications in several contexts when the source field is either vorticity, electric current or magnetic field, from vortex dynamics to astrophysics and plasma physics, where highly braided magnetic fields and currents are present.

We consider a generalization of a one-dimensional stochastic process known in the physical literature as L\'evy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process., Comment: Final version to be published in J. Stat. Phys. 23 pages. (Changes from v1: Theorem 2.4 and Corollary 2.6 have been removed.)

A plane wave is scattered by a potential of bounded support. Translation, rotation and reflection of the potential, q0 induce transformations of the scattered wave. The latter can be represented by means of Born sequences, where q0 appears under the integral sign: non-local formulas are thus derived, the properties of which are discussed. Next, the symmetries induced by the 1 st BORN approximation are addressed. Invariance of the squared modulus of the scattering amplitude holds for translation and reflection. The transformation T e := 1 3 +Σ 3 l=1e l A l, with {e l ;} real and {A l } the generators of rotations in IR 3 , is investigated. Conditions on the {e l } are derived, by which the scattering amplitude coming from the first BORN approximation is invariant to T e . As an application, these “false symmetries” are compared to those induced by limited angular resolution of a detector in light scattering experiments. Namely, scattering patterns are made available by the TAOS (Two-dimensional Angle-resolved Optical Scattering) method, which consists of detecting single airborne aerosol particles and collecting the intensity of the light they scatter from a pulsed, monochromatic laser beam. The optics and the detector properties determine the resolution at which a pattern is saved. The implications on the performance of TAOS pattern analysis are briefly discussed.

We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (ref. [8]), which we also extend to pinning models., Added a heuristic explanation, updated references, and other minor changes; version to appear on CMP

We study diffusion on a periodic billiard table with infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a L\'evy walk combining exponentially-distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [Phys. Rev. Lett. 50, 1959 (1983)]., Comment: 5 pages, 3 figures

We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $ i \partial_t \Psi (t) = H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $\delta$ interaction, with strength $\alpha < 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0m^*$ there is no minimum. Moreover, the set of minimizers has the structure ${\mathcal M}={e^{i\theta}\hat \Psi_m, \theta\in \erre}$. Correspondingly, for every $m, Comment: 26 pages, 1 figure

A plane wave is scattered by a potential of bounded support. Translation, rotation and reflection of the potential, q0 induce symmetries on the scattered wave. The latter is represented by a Born sequence, where the reference potential, q0, appears. The result is applied to derive invariance of the scattering magnitude to finite translation and reflection. Conditions on the entries of an \almost SO(3)" matrix are derived, which induce symmetries of the scattering amplitude by the first Born approximation.

By using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links, and by using ropelength data from the RIDGERUNNER tightening algorithm (Ashton et al 2011 Exp. Math. 20 57–90) we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links. The two spectra are found to follow an almost identical logarithmic law. By assuming that the number of knot types grows exponentially with the topological crossing number, we show that this generic behavior can be justified by a general relationship between ropelength and crossing number, which is in good agreement with former analytical estimates (Buck and Simon 1999 Topol. Appl. 91 245–57, Diao 2003 J. Knot Theory Ramifications 12 1–16). Moreover, by considering the ropelength averaged over a given knot family, we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots.

The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples

We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel'fand-Zakharevich bi-Hamiltonian setting for integrable systems.

We consider the nonlinear Schrodinger equation in dimension one with a nonlinearity concentrated in a finite number of points. Detailed results on the local existence of the solution in fractional Sobolev spaces H ρ are given. We also prove the conservation of the L 2 -norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space H 1 . Finally we prove the existence of blow-up solutions for strongly attractive interaction.

It is shown that a class of Stäckel separable systems is characterized in terms of a Gel'fand-Zakharevich bihamiltonian structure. This structure arises as an extension of a Poisson-Nijenhuis structure on phase space. It is also shown that the Casimir of the Gel'fand-Zakharevich bihamiltonian structure provides the family of commuting Killing tensors found by Benenti and that, because of Eisenhart's theorem, characterize orthogonal separability. It is also shown that recently found properties of quasi-bihamiltonian systems are natural consequences of the geometry of the extension of the Poisson-Nijenhuis structure. © 2000 Elsevier Science B.V

A self consistent gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold is proposed. In the resulting set-up, the “Lagrangian” L is replaced by a section of a suitable principal fiber bundle over the velocity space, called the lagrangian bundle, while the associated Poincaré-Cartan 2-form is recognized as the curvature 2-form of a connection induced by L on a second “co-lagrangian” principal bundle. A parallel construction leads to the identification of a hamiltonian and a co-hamiltonian bundle over the phase space. An analysis of the properties of these spaces provides an intrinsic geometrical characterization of the Legendre transformation, thus allowing a systematic translation of the hamiltonian formalism into the newer scheme.

For pt.1 see ibid., vol.41, no.1, p.34-51 (1999). The Ipswich data provide a unique opportunity for the validation of the approximate back-propagation (ABP) methods, which were originally developed to identify the shape of acoustic scatterers in the resonance region. These methods rely on a heuristic relationship, i.e., ABP, between the expansion coefficients that represent the scattered wave in the far zone and those on the obstacle boundary, /spl Gamma/. The unknown is the shape-parameter vector, /spl psi//spl I.oarr//spl isin//spl Psi//sub ad/, the admissible set. The objective function to be minimized is the L/sup 2/(/spl Gamma/)-norm of the boundary defect. In the vertical-polarization case, ABP consists of an affine map, which is easy to derive. Its ingredients are arrays of inner products in L/sup 2/(/spl Gamma/), where outgoing cylindrical wave functions are involved. The corresponding numerical results, based on the IPS001VV data, are satisfactory. The attraction domain of the expected solution, the reference obstacle (a disk), is numerically determined by varying the initial conditions in a wide subset of /spl Psi//sub ad/. Reconstruction seems to be unique, although no uniqueness condition is known for the obstacle. In the horizontal-polarization case, ABP relies on vector harmonic functions in a cylindrical geometry. The complexity of the algorithm is higher. Results based on the TPS001HH set are summarized. Although the numerical solution does not show any focal minimum other than the reference obstacle, the corresponding attraction domain is smaller than in the vertical-polarization case.

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones.

The elastic scattering of the mass-less (${\mit\omega}^2 = c^2 k^2$) complex scalar field by a potential, $q[.]$, of bounded support is a prototype model which finds applications to acoustics and electromagnetics in a classical (i.e., non quantum mechanical) setting. Of particular interest are the symmetries of the scattered wave, $u[.]$ and of the quantities derived therefrom ($A[.]$, $|A[.]|^2$) caused by symmetry operations on $q[.]$, such as translation, rotation, reflection, and scaling. This investigation is motivated by the analysis of scattering patterns of environmental interest.

In the present paper an introduction to the new subject of nonlinear dispersive hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of nonlinear Schr\"odinger equation. Special consideration is given to existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning NLS equation on a star graph: the standing waves of NLS equation on a graph with a $\delta$ interaction at the vertex; the scattering of fast solitons through an Y-junction in the cubic case. The emphasis is on description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed., Comment: 20 pages; 5 figures. Added reference; corrected typos. Accepted on Phil. Trans. R. Soc. A

The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The rst model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a point (or contact) interaction with strength , which consists of a singular perturbation of the Laplacian described by a selfadjoint operator H , and letting the strength depend on the wave function: i du dt = H u, = (u). It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to jx x0j1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e. the coe cient of its singular part, then, in order to introduce a nonlinearity, we let the strength depend on u according to the law = jqj , with > 0. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form u(t) = ei!t !, which are orbitally stable in the range 2 (0; 1), and orbitally unstable for 1: Moreover, we show that for 2 (0; p1 2 ) [ p1 2 ; p 3+1 2 p 2 every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted Lp space which allows dispersive estimates, the following resolution holds: u(t) = ei!1t+il(t) !1 + Ut 1 + r1, where Ut is the free Schrödinger propagator, !1 > 0 and 1, r1 2 L2(R3) with kr1kL2 = O(tp) as t ! +1, p = 5 4 , 1 4 depending on 2 (0; 1= p 2), 2 (1= p 2; 1), respectively, and nally l(t) is a logarithmic increasing function that appears when 2 (p1 2 ; ), for a certain 2 p1 2 ; p 3+1 2 p 2 i . Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation i du dt = u juj4u. In this case we prove, for any and 0 su ciently small, the existence of radial nite energy solutions of the form u(t; x) = ei (t) 1=2(t)W( (t)x) + ei t + o _H1(1) as t ! +1, where (t) = 0 ln t, (t) = t , W(x) = (1+ 1 3 jxj2)1=2 is the ground state and is arbitrarily small in _H 1.

We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in $H^s$ to the solution of the Hopf equation, provided the initial data belongs to $H^s$, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities., Comment: 23 pages, minor corrections

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrodinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a point (or contact) interaction with strength α, which consists of a singular perturbation of the Laplacian described by a self-adjoint operator Hα, and letting the strength α depend on the wavefunction: iu=Hαu, α = α(u). It is well-known that the elements of the domain of such operator can be written as the sum of a regular function and a function that exhibits a singularity proportional to |x − x0|−1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e., the coefficient of its singular part, then, in order to introduce a nonlinearity, we let the strength α depend on u according to the law α = −ν|q|σ, with ν > 0. This characterizes the model as a focusing NLS (nonlinear Schrodinger) with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form u(t) = eiωtΦω, which are orbitally stable in the range σ ∈ (0, 1), and orbitally unstable when σ ⩾ 1. Moreover, we show that for σ∈(0,12) every standing wave is asymptotically stable in the following sense. Choosing initial data close to the stationary state in the energy norm, and belonging to a natural weighted Lp space which allows dispersive estimates, the following resolution holds: u(t)=eiω∞tΦω∞+Ut*ψ∞+r∞, where U is the free Schrodinger propagator, ω∞ > 0 and ψ∞, r∞∈L2(R3) with ‖r∞‖L2=O(t−5/4) as t→+∞. Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical, in the sense that it does not give rise to blow up, regardless of the chosen initial data.