Nonlocal Perturbation Research Articles

Uniform $$C^{1,\alpha }$$-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter

Uniform $$C^{1,\alpha }$$-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter

Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.

Exact domain truncation for the Morse-ingard equations

Morse and Ingard [1] give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed in [2] for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner. These numerical results taken together combine several advanced techniques, including higher-order finite elements, geometric multigrid in curvilinear geometry, native use of complex arithmetic, and incorporation of nonlocal operators. These are tied together in a high-level simulation using the Firedrake library.

Analytical Representations of Far Fields of Internal Gravity Waves from Local and Nonlocal Perturbation Sources

Analytical Representations of Far Fields of Internal Gravity Waves from Local and Nonlocal Perturbation Sources

Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation

Abstract In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: ( − Δ ) s u = λ u + α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u + ( I μ * ∣ u ∣ 2 μ , s * ) ∣ u ∣ 2 μ , s * − 2 u , in R N , {\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫ R N u 2 d x = c 2 , \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s ∈ ( 0 , 1 ) , N > 2 s , 0 < μ < N , α > 0 , c > 0 s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0 , and I μ ( x ) {I}_{\mu }\left(x) is the Riesz potential given by I μ ( x ) = A μ ∣ x ∣ μ with A μ = Γ μ 2 2 N − μ π N ⁄ 2 Γ N − μ 2 , {I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2 N − μ N < q < 2 μ , s * = 2 N − μ N − 2 s \frac{2N-\mu }{N}\lt q\lt {2}_{\mu ,s}^{* }=\frac{2N-\mu }{N-2s} is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L 2 {L}^{2} -subcritical perturbation α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u \alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u with exponent 2 N − μ N < q < 2 N − μ + 2 s N \frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu +2s}{N} , we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L 2 {L}^{2} -critical and L 2 {L}^{2} -supercritical cases 2 N − μ + 2 s N ≤ q < 2 N − μ N − 2 s \frac{2N-\mu +2s}{N}\le q\lt \frac{2N-\mu }{N-2s} , we also prove that the equation has ground states of mountain-pass-type.

Solving stochastic equations with unbounded nonlinear perturbations

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.

Normalized Solutions to the Critical Choquard-type Equations with Weakly Attractive Potential and Nonlocal Perturbation

Normalized Solutions to the Critical Choquard-type Equations with Weakly Attractive Potential and Nonlocal Perturbation

Observational signatures of circumbinary discs – I. Kinematics

ABSTRACT We present five morphological and kinematic criteria to aid in asserting the binary nature of a protoplanetary disc, based on 3D hydrodynamical simulations of circumbinary discs post-processed with Monte Carlo radiative transfer. We find that circumbinary discs may be identified by (i) a central cavity, (ii) spiral arms both in and outside of their central cavities, (iii) non-localized perturbations in their iso-velocity curves, (iv) asymmetry between the lines of maximum speed of the blueshifted and redshifted wings, and (v) asymmetry between the area of the blueshifted and redshifted wings. We provide quantitative metrics for the last two criteria that can be used, in conjunction with the morphological criteria, to signal whether a protoplanetary disc is likely to be a circumbinary disc.

Practical quantum simulation of small-scale non-Hermitian dynamics

Non-Hermitian quantum systems have recently attracted considerable attention due to their exotic properties. Though many experimental realizations of non-Hermitian systems have been reported, the non-Hermiticity usually resorts to the hard-to-control environments and cannot last for too long times. An alternative approach is to use quantum simulation with the closed system, whereas how to simulate non-Hermitian Hamiltonian dynamics remains a great challenge. To tackle this problem, we propose a protocol which combines a dilation method with the variational quantum algorithm. The dilation method is used to transform a non-Hermitian Hamiltonian into a Hermitian one through an exquisite quantum circuit, while the variational quantum algorithm is for efficiently approximating the complex entangled gates in this circuit. As a demonstration, we apply our protocol to simulate the dynamics of an Ising chain with nonlocal non-Hermitian perturbations, which is an important model to study quantum phase transition at nonzero temperatures. The numerical simulation results are highly consistent with the theoretical predictions, revealing the effectiveness of our protocol. The presented protocol paves the way for practically simulating small-scale non-Hermitian dynamics.

On uniqueness results for solutions of the Benjamin equation

We prove that the uniqueness results obtained in [26] for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With this purpose, we showed that for any solutions u and v defined in R×[0,T], if there exists an open set I⊂R such that u(⋅,0) and v(⋅,0) agree in I, ∂tu(⋅,0) and ∂tv(⋅,0) agree in I, then u≡v. A better version of this uniqueness result is also established. To finish, this type of uniqueness results were also proved for the nonlocal perturbation of the Benjamin-Ono equation (npBO) and for the regularized Benjamin-Ono equation (rBO).

Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato–Lefever waves

In an interesting recent analysis, Haragus–Johnson–Perkins–de Rijk have shown modulational stability under localized perturbations of steady periodic solutions of the Lugiato–Lefever equation (LLE), in the process pointing out a difficulty in obtaining standard “nonlinear damping estimates” on modulated perturbation variables to control regularity of solutions. Here, we point out that in place of standard “inverse-modulated” damping estimates, one can alternatively carry out a damping estimate on the “forward-modulated” perturbation, noting that norms of forward- and inverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classical argument structure of Johnson–Noble–Rodrigues–Zumbrun for parabolic systems. This observation seems of general use in situations of delicate regularity. Applied in the context of (LLE), it gives the stronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.

Normalized solutions to the nonlinear Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent

We study the effect of lower order nonlocal perturbations in the existence of positive solutions to the following nonlinear Choquard equation−Δu=λu+μ(Iα⁎|u|p)|u|p−2u+(Iα⁎|u|2α⁎)|u|2α⁎−2ux∈RN, having prescribed L2−norm∫RN|u|2dx=a>0, where N≥3, α∈(0,N), N+αN<p<2α⁎, μ>0 and λ∈R. Under different assumptions on p, we establish several existence results and characterize the behavior of solutions as μ→0.

Nonlocal pseudospin dynamics in a quantum Ising chain

The existence of topological zero modes in nontrivial phase of quantum Ising chain results in not only the Kramers-like degeneracy spectrum, but also dynamic response for non-Hermitian perturbation in the ordered phase (2021 Phys. Rev. Lett. 126 116 401). In this work, we investigate the possible response of the degeneracy spectrum for Hermitian perturbations. We provide a single-particle description of the model in the ordered phase, associating with an internal degree of freedom characterized as a pseudospin. The effective magnetic field, arising from both local and nonlocal perturbations in terms of string operators, acts on the pseudospin. We show that the action of string operator can be realized via a quench under the local perturbations. As an application, any ground states and excited states for the Hamiltonian with perturbation can be selected to identify the quantum phase, by adding the other perturbations to trigger a quench and measuring the Loschmidt echo.

On nonlinear Miyadera-Voigt perturbations

Let \(A,C,P:D(A)\subset X\rightarrow X\) be linear operators on a Banach space X such that \(-A\) generates a strongly continuous semigroup on X, and \(F:X\rightarrow X\) be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form \({\dot{u}}(t)=G(u(t))\), where \(G:D(A)\rightarrow X\) is a nonlinear map defined by \(G=-A+C+F\circ P\). In fact, using the concept of maximal \(L^p\)-regularity and a fixed point theorem, we establish the existence and uniqueness of a strong solution for the above-mentioned semilinear equation. We illustrate our results by applications to nonlinear heat equations with respect to Dirichlet and Neumann boundary conditions, and a nonlocal unbounded nonlinear perturbation.

Plasma edge density fluctuation measurements via lithium beam emission spectroscopy on EAST

Accurate and stable measurements of edge density fluctuation with high spatio-temporal resolution have been achieved by the lithium beam emission spectroscopy (Li-BES) diagnostic on experimental and advanced superconducting tokamak (EAST). The new narrower band interference filter exhibits good ability to suppress background emission signal even under strong lithium coating of the tokamak. The raw data measured by channels at different spatial locations in avalanche photo diode camera with high chopping frequency show good consistency. Based on the detected experimental data, detailed information of density profile and fluctuation structures is obtained. A clear edge coherent mode in the auto-power spectrum is observed in pedestal region, which is regarded as the dominant factor for the strong pedestal density fluctuation amplitude. The cross-power spectrum analysis further excludes the additional effects of common-mode noises and non-local perturbation, demonstrating that the detected fluctuation is only caused by local density fluctuation. The normalized radial and poloidal wave-number spectra can specify the quantitative changes of radial wavenumber (k r) and poloidal wavenumber (k θ) during the pedestal density fluctuation phase. This new Li-BES system, which can provide more accurate experimental data, allows further studies of edge density fluctuation and complex transport process on EAST.

On the Uniform Stability of Recovering Sine-Type Functions with Asymptotically Separated Zeros

We obtain the uniform stability of recovering entire functions of special form from their zeros. To such a form, we can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, as well as the nonlocal perturbations of these operators. We prove that the dependence of such functions on the sequences of their zeros is Lipschitz continuous with respect to natural metrics on each ball of a finite radius. Results of this type can be used for studying the uniform stability of inverse spectral problems. In addition, general theorems on the asymptotics of zeros of functions of this class and on their equivalent representation via an infinite product are obtained, which give the corresponding results for many specific operators.

Cubic-quartic optical solitons in couplers with optical metamaterials having parabolic non-local law nonlinearity

The main goal of this article is to investigate the optical soliton solutions for the cubic–quartic (CQ) twin-core couplers and multiple-core couplers having optical metamaterials with parabolic non-local law nonlinearity and nonlinear perturbation terms. The couplings with nearest neighbors and with all neighbors are discussed. The unified Riccati equation expansion approach and the addendum to Kudryashov’s approach are applied. Straddled, combo-bright-singular, dark, bright, singular solitons solutions as long as periodic and rational solutions are found.

Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires

Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. This result does not contradict the bulk-edge correspondence: indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.

The fragmentation equation with size diffusion: Well posedness and long-term behaviour

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Nonlinear differential equations with perturbed Dirichlet integral boundary conditions

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green’s function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.