Irregular Solutions Research Articles

Solving differential equations using deep neural networks

Recent work on solving partial differential equations (PDEs) with deep neural networks (DNNs) is presented. The paper reviews and extends some of these methods while carefully analyzing a fundamental feature in numerical PDEs and nonlinear analysis: irregular solutions. First, the Sod shock tube solution to the compressible Euler equations is discussed and analyzed. This analysis includes a comparison of a DNN-based approach with conventional finite element and finite volume methods, and demonstrates that the DNN is competitive in terms of degrees of freedom required for a given accuracy. Further, the DNN-based approach is extended to consider performance improvements and simultaneous parameter space exploration. Next, a shock solution to compressible magnetohydrodynamics (MHD) is solved for, and used in a scenario where experimental data is utilized to enhance a PDE system that is a priori insufficient to validate against the observed/experimental data. This is accomplished by enriching the model PDE system with source terms that are then inferred via supervised training with synthetic experimental data. The resulting DNN framework for PDEs enables straightforward system prototyping and natural integration of large data sets (be they synthetic or experimental), all while simultaneously enabling single-pass exploration of an entire parameter space.

Existence of solutions with a horizon in pure scalar-Gauss-Bonnet theories

We consider the Einstein-scalar-Gauss-Bonnet theory and assume that, at regimes of large curvature, the Ricci scalar may be ignored compared to the quadratic Gauss-Bonnet term. We then look for static, spherically-symmetric, regular black-hole solutions with a non-trivial scalar field. Despite the use of a general form of the spacetime line-element, no black-hole solutions are found. In contrast, solutions that resemble irregular particle-like solutions or completely regular gravitational solutions with a finite energy-momentum tensor do emerge. In addition, in the presence of a cosmological constant, solutions with a horizon also emerge, however, the latter corresponds to a cosmological rather than to a black-hole horizon. It is found that, whereas the Ricci term works towards the formation of the positively-curved topology of a black-hole horizon, the Gauss-Bonnet term exerts a repulsive force that hinders the formation of the black hole. Therefore, a pure scalar-Gauss-Bonnet theory cannot sustain any black-hole solutions. However, it could give rise to interesting cosmological or particle-like solutions where the Ricci scalar plays a less fundamental role.

Scattering with Manning–Rosen potential in all partial waves

Ordinary differential approach together with certain properties of Gaussian hypergeometric functions is exploited to construct exact analytical expressions for regular and irregular solutions in all partial waves for motion in Manning–Rosen potential. The Jost function thus obtained from the near the origin behaviour of the irregular solution is used appropriately to compute scattering phase shifts for the neutron–proton and neutron–deuteron systems for few lower partial waves. Our results are in good agreement with experimental data.

Stellar Dynamos in the Transition Regime: Multiple Dynamo Modes and Antisolar Differential Rotation

Abstract Global and semi-global convective dynamo simulations of solar-like stars are known to show a transition from an antisolar (fast poles, slow equator) to solar-like (fast equator, slow poles) differential rotation (DR) for increasing rotation rate. The dynamo solutions in the latter regime can exhibit regular cyclic modes, whereas in the former one, only stationary or temporally irregular solutions have been obtained so far. In this paper we present a semi-global dynamo simulation in the transition region, exhibiting two coexisting dynamo modes, a cyclic and a stationary one, both being dynamically significant. We seek to understand how such a dynamo is driven by analyzing the large-scale flow properties (DR and meridional circulation) together with the turbulent transport coefficients obtained with the test-field method. Neither an αΩ dynamo wave nor an advection-dominated dynamo are able to explain the cycle period and the propagation direction of the mean magnetic field. Furthermore, we find that the α effect is comparable or even larger than the Ω effect in generating the toroidal magnetic field, and therefore, the dynamo seems to be of α 2Ω or α 2 type. We further find that the effective large-scale flows are significantly altered by turbulent pumping.

Regular and Jost states for the S-wave Manning–Rosen potential

A regular solution for the Manning–Rosen potential is constructed by adapting the differential equation approach to the problem. The Jost solution and the Jost function are found by exploiting the relation between regular and irregular solutions. The Jost function thus obtained is applied for the first time to find bound state energies and the scattering phase shifts for nuclear systems.

Analytical S-matrix derivation in the theory of the nonhydrogenic Stark effect

An analytical scattering matrix is found for Rydberg atoms in the external electric field at negative energies. The exact relation between the Coulomb spherical and parabolic irregular solutions is obtained, which is then used to find the physical solution of the Stark problem bounded at infinity and satisfying the quantum-defect boundary condition within the region next to the ion core where the external field does not mix the states with a definite orbital momentum. The S matrix thus found can be used for interpreting various experimental data, such as positions and widths of lines in the photo-absorption spectra, photoionization cross-section, and ionization microscopy.

The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

Abstract We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue L p {L^{p}} -space, 1 < p < ∞ {1<p<\infty} . The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}} , and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

Abstract While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ⁢ ( Ω ) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} , 1 < q < ∞ {1<q<{\infty}} , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W 0 1 , q ⁢ ( Ω ) {W^{1,q}_{0}(\Omega)} - W 0 1 , q ′ ⁢ ( Ω ) {W_{0}^{1,q^{\prime}}(\Omega)} functional setting, 1 q + 1 q ′ = 1 {\frac{1}{q}+\frac{1}{q^{\prime}}=1} . The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of W 1 , q ′ {W^{1,q^{\prime}}} -stability of the H 0 1 {H_{0}^{1}} -projector.

Stable coherent terahertz synchrotron radiation from controlled relativistic electron bunches

Spontaneous formation of spatial structures (patterns) occurs in various contexts, ranging from sand dunes and rogue wave formation, to traffic jams. These last decades, very practical reasons also led to studies of pattern formation in relativistic electron bunches used in synchrotron radiation light sources. As the main motivation, the patterns which spontaneously appear during an instability increase the terahertz radiation power by factors exceeding 10000. However the irregularity of these patterns largely prevented applications of this powerful source. Here we show how to make the spatio-temporal patterns regular (and thus the emitted THz power) using a point of view borrowed from chaos control theory. Mathematically, regular unstable solutions are expected to coexist with the undesired irregular solutions, and may thus be controllable using feedback control. We demonstrate the stabilization of such regular solutions in the Synchrotron SOLEIL storage ring. Operation of these "controlled unstable solutions" enables new designs of high charge and stable synchrotron radiation sources.

STRONGLY IRREGULAR PERIODIC SOLUTIONS OF THE FIRST-ORDER LINEAR HOMOGENEOUS DISCRETE EQUATION

In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.

Regular and irregular solutions in the problem of dislocations in solids

Рассмотрены асимптотические по невязке решения исходного дифференциального уравнения с отклонениями пространственной переменной. Все решения естественным образом разбиваются на классы, регулярно и нерегулярно зависящие от параметров задачи. В различных областях из малой окрестности нулевого состояния равновесия фазового пространства построены специальные нелинейные распределенные уравнения и системы уравнений, зависящие от континуальных семейств некоторых параметров. В частности, показано, что решения исходного пространственного одномерного уравнения можно описать решениями специальных уравнений и систем уравнений шредингеровского типа в пространственно двумерной области изменения аргументов.

Regular and Irregular Solutions in the Problem of Dislocations in Solids

For an initial differential equation with deviations of the spatial variable, we consider asymptotic solutions with respect to the residual. All solutions are naturally divided into classes depending regularly and irregularly on the problem parameters. In different regions in a small neighborhood of the zero equilibrium state of the phase space, we construct special nonlinear distribution equations and systems of equations depending on continuous families of certain parameters. In particular, we show that solutions of the initial spatially one-dimensional equation can be described using solutions of special equations and systems of Schr¨odinger-type equations in a spatially two-dimensional argument range.

A functional equation with Borel summable solutions and irregular singular solutions

A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.

The effect of closed channels on the electron impact excitation of Mg+, Cd+ ions

Based on the developed method for solving the multi-channel equation, which had been applied to the calculations of several kinds of ions including only open-open interactions, closed channels and their interactions with open channels have been studied. The wave functions of the closed channels are also expressed in terms of their homogeneous solutions which is just the same as for open channels. The homogeneous solutions are described and solved in WKB form, therefore the regular and irregular solutions as well as the quantum defect numbers can be obtained simultaneously. Excitations of Mg+, Cd+ ions impact by electrons are calculated for energies close to the thresholds. The results are compared with those of the experimental observations and previous theoretical calculations. The effect of including the closed channels, especially when the energy passes through the resonance energies, has been discussed according to the deduced formulae and the calculated results.

Structure and Cathodoluminescent Properties of Y2O3:Eu Thin Films at Different Activator Concentrations

The surface structure and cathodoluminescence (CL) spectra of thin films of Y2O3:Eu obtained by HF ion plasma sputtering were investigated by varying the activator concentration in the range of 1.0–7.5 mole%. The possibility of creating irregular solutions of yttrium and europium oxides and the structural features of the small and large crystallites that form the Y2O3:Eu film were demonstrated on the basis of the shape of the CL spectra at different activator concentrations. The critical distance for interaction between the Eu3+ ions at nodes with C2 and C3i point symmetry was estimated, and a mechanism of energy migration between these centers was proposed.

Fully-relativistic full-potential multiple scattering theory: A pathology-free scheme

The Green function plays an essential role in the Korringa–Kohn–Rostoker(KKR) multiple scattering method. In practice, it is constructed from the regular and irregular solutions of the local Kohn–Sham equation and robust methods exist for spherical potentials. However, when applied to a non-spherical potential, numerical errors from the irregular solutions give rise to pathological behaviors of the charge density at small radius. Here we present a full-potential implementation of the fully-relativistic KKR method to perform ab initio self-consistent calculation by directly solving the Dirac differential equations using the generalized variable phase (sine and cosine matrices) formalism Liu et al. (2016). The pathology around the origin is completely eliminated by carrying out the energy integration of the single-site Green function along the real axis. By using an efficient pole-searching technique to identify the zeros of the well-behaved Jost matrices, we demonstrated that this scheme is numerically stable and computationally efficient, with speed comparable to the conventional contour energy integration method, while free of the pathology problem of the charge density. As an application, this method is utilized to investigate the crystal structures of polonium and their bulk properties, which is challenging for a conventional real-energy scheme. The noble metals are also calculated, both as a test of our method and to study the relativistic effects.

A posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

We propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

Application of reconstitution multiple scale asymptotics for a two-to-one internal resonance in Magnetic Resonance Force Microscopy

In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content.

Influence of Obtainment Conditions and Excitation Into Spectral and Kinetic Characteristics of Cathodoluminescence in Y2O3:Eu Thin Films

The spectra and kinetics of the rise and decay of the cathodoluminescence (CL) of thin films of Y2O3:Eu obtained by RF-magnetron sputtering were investigated. Based on the shape of the CL spectra at different excitation energies, it showed the possibility of creating irregular solutions of yttrium and europium oxides and the structural features of the surface and bulk layers. The time constant for the decay CL for 612 nm emission was determined and the value of which is within the range (1.8 - 4.1) ms. This value is a complex function of the type of film deposition atmosphere, the activator concentration, and the duration of the exciting pulses was shown. The features of the risen CL and the proposal based on the delayed rise CL analyze structural perfection of Y2O3:Eu thin films.

Scattering and Bound States of a Spin-1/2 Neutral Particle in the Cosmic String Spacetime

In this paper the relativistic quantum dynamics of a spin-1/2 neutral particle with a magnetic moment μ in the cosmic string spacetime is reexamined by applying the von Neumann theory of self-adjoint extensions. Contrary to previous studies where the interaction between the spin and the line of charge is neglected, here we consider its effects. This interaction gives rise to a point interaction: ∇·E=(2λ/α)δ(r)/r. Due to the presence of the Dirac delta function, by applying an appropriated boundary condition provided by the theory of self-adjoint extensions, irregular solutions for the Hamiltonian are allowed. We address the scattering problem obtaining the phase shift, S-matrix, and the scattering amplitude. The scattering amplitude obtained shows a dependency with energy which stems from the fact that the helicity is not conserved in this system. Examining the poles of the S-matrix we obtain an expression for the bound states. The presence of bound states for this system has not been discussed before in the literature.