131,835 research outputs found
La Grande Guerra dall'antica Grecia alla Danimarca: "Spartanerne" (1919) di Emil Bønnelycke
No full textUn'esperienza mi(s)tica: i viaggi di Hamsun e di Rilke in Russia all'alba del ventesimo secolo
No full textIn the late 19th and early 20th century, Russia and the Slavic world were considered by some Western poets and intellectuals as a part of Europe where human society had not (yet) been spoiled by the industrial revolution, and whose people, therefore, kept more primitive, genuine and vital values and customs alive. Russia was visited by the Norwegian Knut Hamsun in 1899 and by the German-speaking Rainer Maria Rilke in 1899 and 1900, and their experiences are recorded in significant works and letters. This paper examines aspects of their travels and highlights similarities and differences in their relationship with the places, peoples and cultures they described, pointing out Hamsun’s mythical approach and Rilke’s mystical attitude
The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit
No full textIn the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:(Formula Presented.)(Formula Presented.) This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit (Formula Presented.) the weak (integral) dynamics converges in (Formula Presented.) to the weak dynamics of the NLS with point-concentrated nonlinearity: (Formula Presented.) where Hα is the Laplacian with the nonlinear boundary condition at the origin (Formula Presented.) and (Formula Presented.). The convergence occurs for every (Formula Presented.) if V ≥ 0 and for every (Formula Presented.) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points
Stationary states of NLS on star graphs
No full textWe define a nonlinear Schr\"odinger equation (NLS) with a power nonlinearity of focusing type on the ramified structure given by edges connected at a vertex with boundary conditions generalizing the potential of strenght on the line, including as a special case () the free propagation. We show that nonlinear stationary states exist both for attractive ( and repulsive ( interaction and we give explicitly their expression. In the case of attractive interaction at the vertex and nonlinearity we characterize the ground state as minimizer of a constrained action and we rigorously discuss its orbital stability. Finally we show that in the free case, for even only, the stationary states can be used to construct traveling waves on the grap
Well posedness of the nonlinear Schrödinger equation with isolated singularities
No full textWe study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers
On the structure of critical energy levels for the cubic focusing NLS on star graphs
No full textWe provide information on a non-trivial structure of phase space of the cubic nonlinear Schrödinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrödinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L2-norm. We moreover show that the only stationary state with prescribed L2-norm is indeed a saddle point
Constrained energy minimization and orbital stability for the NLS equation on a star graph
No full textOn a star graph script G, we consider a nonlinear Schrödinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i∂tψ(t)= -Δψ(t) - |ψ(t)|2μψ(t) + αδ0ψ(t), where the strength α of the vertex interaction is negative and the wave function ψ is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0 < μ ≤ 2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m∗ it attains its minimum value at a certain ψm ∈H1(script G). Moreover, the set of minimizers has the structure script M = {eiθψm, θ ∈ double-struck R}. Correspondingly, for every m < m∗ there exists a unique ω = ω(m) such that the standing wave ψωeiωt is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to α =
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