Nonlinear Schrödinger equation with a point defect

The Nordstr\om-Vlasov system is a relativistic Lorentz invariant generalization of the Vlasov-Poisson system in the gravitational case. The asymptotic behavior of solutions and the non-linear stability of steady states are investigated. It is shown that solutions of the Nordstr\om-Vlasov system with energy grater or equal to the mass satisfy a dispersion estimate in terms of the conformal energy. When the energy is smaller than the mass, we prove existence and non-linear (orbital) stability of a class of static solutions (isotropic polytropes) against general perturbations. The proof of orbital stability is based on a variational problem associated to the minimization of the energy functional under suitable constraints.

The Maxwell–Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension d=3 the system is charge-critical, that is, L^2-critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions d=1,2, global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if d=3), by demonstrating ill-posedness below the charge regularity. In fact, for d le 3 we exhibit a spinor datum belonging to H^s(mathbb {R}^d) for s<0, and to L^p(mathbb {R}^d) for 1 le p < 2, but not to L^2(mathbb {R}^d), which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\geq 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\geq 5$, and a careful study of the nonlinear structure of the quadratic terms in dimension $3$ and $4$ involving the theory of space time resonance.

Experimental high T c systems are intrinsically strongly disordered by point defects. Here we consider two possible consequences of this fact: 1) The magnetic phases of prominent models for high T c are strongly affected. We present numerical calculations of the phase diagram of the Hubbard model as a function of the concentration of point defects and of the band filling in the Hartree Fock approximation. The results may explain the failure to experimentally observe a spin density wave phase predicted by Hartree Fock calculations without defects. 2) The superconductivity of the system may be strongly affected. We consider a model in which the pairing interactions are generated by point defects. We present numerical data on the density of states, T c and local charge density in the BCS approximation, while taking full account of the disorder. The stability of the BCS state was also checked using a variational Monte Carlo calculation.

Orbital stability in the cubic defocusing NLS equation: II. The black soliton

We present new extended Strichartz estimates for the solutions to the heat equation with inhomogeneous nonlinearity in the mass-energy intercritical framework in space dimension d ≥ 1. As an application we show local and global well-posedness in the Strichartz space Lq((0,T); Lr(ℝd)), both in the focusing and the defocusing case, assuming the initial data are in the Sobolev space Hσ(ℝd)), with σ ∈ (0, 1].

To study the stability of a nominal cyclic steady state in power electronic converters, it is necessary to obtain a linearization around the periodic orbit. In many past studies, this was achieved by explicitly deriving the Poincare map that describes the evolution of the state from one clock instant to the next and then locally linearizing the map at the fixed point. However, in many converters, the map cannot be derived in closed form, and therefore this approach cannot directly be applied. Alternatively, the orbital stability can be worked out by studying the evolution of perturbations about a nominal periodic orbit, and some studies along this line have also been reported. In this paper, we show that Filippov's method - which has commonly been applied to mechanical switching systems - can be used fruitfully in power electronic circuits to achieve the same end by describing the behavior of the system during the switchings. By combining this and the Floquet theory, it is possible to describe the stability of power electronic converters. We demonstrate the method using the example of a voltage-mode-controlled buck converter operating in continuous conduction mode. We find that the stability of a converter is strongly dependent upon the so-called saltation matrix - the state transition matrix relating the state just after the switching to that just before. We show that the Filippov approach, especially the structure of the saltation matrix, offers some additional insights on issues related to the stability of the orbit, like the recent observation that coupling with spurious signals coming from the environment causes intermittent subharmonic windows. Based on this approach, we also propose a new controller that can significantly extend the parameter range for nominal period-1 operation.

The Euler-Korreweg model results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, which takes into account capillarity effects in regions where the density experiences large variations, typically across interfaces for fluids exhibiting phase changes. One of the main difficulties in the analysis of the Cauchy problem for this model, a third order system of conservation laws, is the absence of dissipative regularization, since viscosity is neglected. The Cauchy problem for isothermal fluids in one space dimension has been addressed by the authors in an earlier paper, using Lagrangian coordinates. Here the Cauchy problem is investigated in arbitrary space dimension N, still for isothermal fluids, and a variable capillarity coefficient. A local well-posedness result is obtained in Sobolev spaces as though the density gradient and the velocity field were solutions of a symmetrizable hyperbolic system. More precisely, well-posedness is shown for Hs+1 x H-s (s > N/2 + 1) perturbations of smooth global solutions, either constant states or traveling profiles. In addition, almost-global existence is proved for small enough perturbations, and a blow-up criterion is shown. Proofs rely on a suitable extended formulation of the system, which turns out to amount to a nonlinear degenerate Schrodinger equation coupled with a transport equation, and on a priori estimates without loss of derivatives for the extended system, which necessitate various 'gauge' functions to cancel out bad commutators.

Stability and related properties of vacua and ground states

Based on the first-principle calculations by using the Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) method in connection with the local density approximation (LDA), we study theoretically the electronic and magnetic properties of different point defects in ZnO, which are Zinc interstitials (Zni), Zinc antisites (ZnO), Oxygen interstitials (Oi) and Oxygen antisites (OZn) defects in ZnO. The supercell calculations were also performed using the full potential local-orbital (FPLO) band structure scheme. This work presents detailed information about total and local density of states at some concentrations of these defects; the stability of the ferromagnetic state compared with the spin-glass state is investigated by comparing calculating their total energy. The results show on one hand that Zni and ZnO produce a shallow donor bellow the bottom of the conduction band (CB), while Oi and OZn produces the shallow acceptors above the top of the valence band (VB), and moment magnetic; on other hand that the ferromagnetic state is more stable than the spin-glass in Oxygen interstitials (Oi) and vice versa for oxygen antisites (OZn) of native point defects in ZnO. The other native point defects (Zni, ZnO, VO, and VZn) have a zero magnetic moment. The results show that the Curie temperature increases with the concentration of interstitial oxygen.

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces $ \widehat{H}^{s,r} $, where $ \|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}} $, $ \frac{1}{r}+\frac{1}{r'} = 1 $. The assumed regularity for the data is almost optimal with respect to scaling as $ r \to 1 $. This closes the gap between what is known in the case $ r = 2 $, namely $ s &gt; \frac{3}{4} $, and the critical value $ s_c = \frac{1}{2} $ with respect to scaling.

Long time dynamics of Nernst-Planck-Navier-Stokes systems

This research deals with the synthesis of control law by means of Analytic Programming (AP) for the Hénon Map, which is discrete chaotic system. The tool for symbolic regression (AP) is used for the purpose of stabilization of stable state and higher periodic orbits, which represent oscillations between several values of chaotic system. For experimentation, Self‐Organizing Migrating Algorithm (SOMA) with AP and Differential Evolution (DE) as the second algorithm for meta‐evolution were used.

We report on several improvements of the classical time-delayed feedback control method for stabilization of unstable periodic orbits or steady states. In an electronic circuit experiment, we were able to realize time-varying and distributed delays in the control force leading to successful control for large parameter sets, including large time delays. The presented techniques make advanced use of the natural torsion of the orbits, which is also necessary for the original control method to work.

The Cauchy problem for the Maxwell-Klein-Gordon equations in the Lorenz gauge in n space dimensions (n ≥ 4) is shown to be locally well-posed for low regularity (large) data. The result relies on the null structure for the main bilinear terms, which was shown to be present not only in the Coulomb gauge but also in the Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces. Crucial for the improvement are the solution spaces introduced by Klainerman-Selberg.