Space H1 Research Articles

Yang–Mills solutions on Minkowski space via non-compact coset spaces

We find a two-parameter family of solutions of the Yang–Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space H3≅SO(1,3)/SO(3), while the exterior of the lightcone employs de Sitter space dS≅3SO(1,3)/SO(1,2). The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang–Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang–Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.

Fricke Topological Qubits

We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 512, the Berge link 622 or the double-eight link 632) or the link 732, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

The nonlinear Schrödinger equationon the half-line with homogeneous Robin boundary conditions.

We consider the nonlinear Schrödinger equationon the half-line with a Robin boundary condition at and with initial data in the weighted Sobolev space . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of theequation.

Nonlinear stability of multi-solitons for the Hirota equation

In this paper, we study the nonlinear stability of solitons and multi-solitons for the following Hirota equationiut+uxx+2|u|2u+iα(uxxx+6|u|2ux)=0,(t,x)∈R2,α∈R. Firstly, by employing the variational approach with suitable conservation laws, we establish the orbital stability of the Hasimoto solitons in Sobolev space H1(R). Secondly, by choosing a suitable Lyapunov functional and investigating the spectrum of second variation operator around the solitons, we derive that the Hasimoto solitons are also orbitally stable in Hm(R) with m∈N for any α∈R. Finally, pure N-solitons of the Hirota equation are shown to be dynamically stable in HN(R). By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a 2N order elliptic equation, and the dynamical stability issue is reduced to study of the spectrum of explicit linearized systems. Our approach in the spectral analysis consists in an invariant for the multi-solitons and new operator identities motivated by recursion operators of the Hirota equation. As a direct consequence, we recover a new proof of the stability of N-solitons for the nonlinear Schrödinger equation [24].

Chern-Simons limit of ground state solutions for the Schrödinger equations coupled with a neutral scalar field

We consider a class of the Schrödinger equation coupled with an neutral scalar field in non-radial symmetric space H1(R2). In this paper, combining the Nehari manifold, Moser iteration and some analytical skills, we get a positive ground state solution to the problem, which is classical and spherically symmetric. At the same time, this solution together with its derivatives up to order 2 has exponential decay at infinity. Moreover, the asymptotic behavior of solutions is analyzed under the sense of Chern-Simons limit. Our work can be regarded as a supplement and extension to Han et al. (2014) [12], where they rely heavily on the symmetry of work space.

Strong attractors for weakly damped quintic wave equation in bounded domains

In this paper, we study the longtime dynamics for the weakly damped wave equation with quintic non-linearity in a bounded smooth domain of R3. Based on the Strichartz estimates for the case of bounded domains, we establish the existence of a strong global attractor in the phase space H2(Ω)∩H01(Ω)×H01(Ω). Moreover, the finite fractal dimension of the attractor is also shown with the help of the quasi-stable estimation.

Posinormal composition operators on H2

A bounded linear operator A on a Hilbert space is posinormal if there exists a positive operator P such that AA⁎=A⁎PA. Posinormality of A is equivalent to the inclusion of the range of A in the range of its adjoint A⁎. Every hyponormal operator is posinormal, as is every invertible operator. We characterize both the posinormal and coposinormal composition operators Cφ on the Hardy space H2 of the open unit disk D when φ is a linear-fractional selfmap of D. Our work reveals that there are composition operators that are both posinormal and coposinormal yet have powers that fail to be posinormal.

On the cohomological equation of a linear contraction

In this paper, we study the discrete cohomological equation of a contracting linear automorphism A of the Euclidean space Rd. More precisely, if δ is the cobord operator defined on the Fréchet space E = Cl (Rd) (0 ≤ l ≤ ∞) by: δ(h) = h − h ◦ A, we show that: • If E = C0(Rd), the range δ (E) of δ has infinite codimension and its closure is the hyperplane E0 consisting of the elements of E vanishing at 0. Consequently, H1 (A, E) is infinite dimensional non Hausdorff topological vector space and then the automorphism A is not cohomologically C0-stable. • If E = Cl (Rd), with 1 ≤ l ≤ ∞, the space δ (E) coincides with the closed hyperplane E0. Consequently, the cohomology space H1 (A, E) is of dimension 1 and the automorphism A is cohomologically Cl-stable.

Sequences of [formula omitted]-term deviations in Hilbert space

Let D be a dictionary in a Hilbert space H, that is, a set of unit elements whose linear combinations are dense in H. We consider the least m-term deviation σm(x) of an element x∈H: this is the distance from x to the set of all m-term linear combinations of elements of D. We prove a dichotomy result: for any dictionary D, either the sequence {σm(x)}m=0∞ decreases exponentially for every x∈H, or the rate of convergence σm(x)→0 can be arbitrarily slow. We seek universal dictionaries realizing all strictly decreasing null sequences as sequences of m-term deviations. All commonly used dictionaries turn out not to be universal. In particular, the least rational deviations in Hardy space H2 do not form certain strictly monotone null sequences. There are no universal dictionaries in finite dimensional Hilbert spaces. We construct a universal dictionary in every infinite dimensional Hilbert space.

Stability result for the surface quasi-geostropic equations with horizontal dissipation in anisotropic Sobolev space

The stability of smooth solutions of an anisotropic surface quasi-geostrophic equation with horizontal dissipation remains an open problem. In this work, we present a partial answer to this problem in a rougher function space. More precisely, if the initial data θ0 belong to anisotropic Sobolev space H0,s with 12<s<1, then there exists a global small solution, provided that the initial data are small in H0,s. The stability problem in the space H0,1 or more regular spaces remains open. The main tools used are the Littlewood–Paley theory and standard techniques.

Motion planing for an ensemble of Bloch equations towards the south pole with smooth bounded control

One considers the control problem of an ensemble of Bloch equations (non-interacting half-spins) in a static magnetic field B0. The state M(t,⋅) belongs to the Sobolev space H1((ω∗,ω∗),S2) where the parameter ω∈(ω∗,ω∗) is the Larmor frequency. Previous works have constructed a Lyapunov based stabilizing feedback in a convenient H1-norm that assures local L∞-convergence of the initial state M0(ω) to the south pole, solving locally the approximate steering problem from M0 towards the south pole. However, the corresponding control law contains a comb of periodic π-Rabi pulses (Dirac impulses), corresponding to strongly unbounded control. The present work proposes smooth uniformly bounded time-varying controls for this local steering problem, where the Rabi pulses are replaced by adiabatic following smooth pulses. Furthermore, simulations show that this new strategy produces faster convergence, even for initial conditions “relatively far” from the south pole.

2-Complex Symmetric Composition Operators on H2

In this paper, we study 2-complex symmetric composition operators with the conjugation J, defined by Jf(z)=(f(z¯))¯, on the Hardy space H2. More precisely, we obtain the necessary and sufficient condition for the composition operator Cϕ to be 2-complex symmetric with J when ϕ is an automorphism of D. We also characterize 2-complex symmetric with J when ϕ is a linear fractional self-map of D.

Non existence and strong ill-posedness in Ck and Sobolev spaces for SQG

We construct solutions in R2 with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in Ck (k≥2) but that are not in Ck for t>0. We prove a similar result also for Hs in the range s∈(32,2). Moreover, we prove strong ill-posedness in the critical space H2.

Selective CO2 reduction to methane catalyzed by mesoporous Ru-Fe3O4/CeOx-SiO2 in a fixed bed flow reactor

Development of a robust catalyst for selective reduction of CO2 directly into hydrocarbon fuels is very challenging from both energy and environmental perspectives as it offers a renewable and green route for the production of fuels. Herein, we report a novel catalyst synthesized via excess solution impregnation of Ru and Fe3O4 nanoparticles (NPs) on ceria promoted mesoporous silica SBA-15 support. The selective catalytic reduction of CO2 to methane was carried out in the presence of H2 over novel Ru-Fe3O4/CeOx-SiO2 (Ce3+/Ce4+, x=1.64) catalyst in a fixed bed reactor. The CO2 conversion was found to be 82% at 0.25 wt% ruthenium loading, 2.5 wt% iron loading, 575 K temperature, 20 bar pressure, 3000 mLg−1h−1 gas hour space velocity and H2 to CO2 mole ratio of 5:1. The close contact between ruthenium and Fe3O4 nanoparticles facilitated the reduction of CO2 through hydrogen spill-over effect at lower temperature, whereas ceria NPs acted as a promoter for this reduction reaction. The catalysts were characterized thoroughly using physicochemical techniques such as CO chemisorption, BET, TPR, TPD, XRD, SEM, HR- TEM, ICP-AES and XPS analyses. High surface area and large mesopores of silica support facilitated the fine dispersion of the active catalytic sites and oxygen vacancy as supported from the DFT study on the catalytic activity. Optimal process conditions could render much higher CO2 conversion efficacy for selective methane synthesis in comparison with previous investigations.

Global existence of unique solutions to equations for pattern formation in active mixtures

In this paper, we deal with two models for pattern formation in active system on the d-dimensional torus Td=−ππd, d ≥ 2, with the periodic boundary conditions.(1)We first consider the model in (Lee and Kardar, 2001) describing the density and the tubule orientation field. After perturbing the orientation field around (1,0), we show that there is a unique global-in-time solution to the perturbed model when initial data is sufficiently small in the energy space H2.(2)The second model under consideration is Active model C in (Maryshev et al., 2020). In this case, we perturb the density around 12, which generates a damping term, and we prove that there is a unique global-in-time solution when initial data is sufficiently small in Wiener space A0.Although these two models are treated in different ways, both models are presented in this paper from the perspective of dealing with pattern formation phenomenon.

Compensated compactness: Continuity in optimal weak topologies

For l-homogeneous linear differential operators A of constant rank, we study the implicationvj⇀vinXAvj→AvinW−lY}⇒F(vj)⇝F(v)inZ, where F is an A-quasiaffine function and ⇝ denotes an appropriate type of weak convergence. Here Z is a local L1-type space, either the space M of measures, or L1, or the Hardy space H1; X,Y are Lp-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of X,Y,Z are sharp. Analogous statements are also given in the case when F(v) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove Hp-bounds for the sequence (F(vj))j, for appropriate p<1, and new convergence results in the dual of Hölder spaces when (vj) is A-free and lies in a suitable negative order Sobolev space W−β,s. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.

Composition operators on Hardy-Smirnov spaces

We investigate composition operators CΦ on the Hardy-Smirnov space H2(Ω) induced by analytic self-maps Φ of an open simply connected proper subset Ω of the complex plane. When the Riemann map τ:U→Ω used to define the norm of H2(Ω) is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on H2(Ω). These descriptions are particular cases of the results obtained by Gunatillake-Jovovic-Smith in 2015 and Gunatillake in 2017. Additionally, if the coefficients of τ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on H2(Ω). We finish this paper showing that if Ω is unbounded and Φ is a non-automorphic self-map of Ω with a fixed point, then CΦ is never complex symmetric on H2(Ω).

Some remarks on the inhomogeneous biharmonic NLS equation

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation iut+Δ2u+λ|x|−b|u|αu=0,where λ=±1 and α, b>0. In the subctritical case, we improve the global well-posedness result obtained in Guzmán and Pastor (2020) for dimensions N=5,6,7 in the Sobolev space H2(RN). The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, α=8−2bN−4 are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space Ḣ2.

Another look at planar Schrödinger-Newton systems

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth{−Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2, where f is a smooth reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space H1(R2). The analysis developed in this paper also allows to investigate the relationship between a Schrödinger-Newton system of Riesz-type and a Schrödinger-Newton system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrödinger-Newton system and may it have some potential applications in various related problems.

Tensor rank bounds for point singularities in ℝ3

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ε ∈ (0,1) in the Sobolev space H1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q = (0,1)3. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.