Degenerate Case Research Articles

Steady secondary flow in a turbulent boundary layer past a slender axisymmetric body

The turbulent boundary layer in a viscous incompressible fluid developing longitudinally past the surface of a thin cone or cylinder at a finite distance from the laminar–turbulent transition zone is studied. The characteristic Reynolds number, determined from the external flow velocity and the length of the body, is assumed to be large, and the thickness of the boundary layer is small and comparable to the radius of the body. The asymptotic method of multiple scales is used to find solutions to the Navier–Stokes equations. Instead of the traditional decomposition of the solution into time-averaged values and their fluctuations, the velocities and pressure are expressed as an asymptotic series consisting of steady and perturbed terms. As a result, the viscous steady flow (‘secondary’) that arises in the boundary layer as a mandatory component of fast turbulent fluctuations was described. Analytical and numerical solutions for the radial steady velocity are presented, describing the self-induced suction of fluid from the external flow into the boundary layer. Further analytical solutions are obtained for the longitudinal and circumferential velocities, which differ markedly from the laminar regime. The solutions found are somewhat similar to the degenerate (one-dimensional) case of self-sustaining longitudinal thin structures in turbulent shear flows. A qualitative comparison with direct numerical simulations is presented.

On the existence and estimates of inverse functions in the degenerate case

On the existence and estimates of inverse functions in the degenerate case

Ohta–Kawasaki energy for amphiphiles: Asymptotics and phase-field simulations

We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.

Authentic Majorana versus singlet Dirac neutrino contributions to μ+μ+→ℓ+ℓ+ (ℓ=e,τ) transitions

We revisited the computation of the cross section for the μ+μ+→ℓ+ℓ+ (ℓ+=τ+,e+) transitions in the presence of new heavy Majorana neutrinos. We focus on scenarios where the masses of the new states are around some few TeV, the so-called low-scale seesaw models. Our analysis is set in a simple model with only two new heavy Majorana neutrinos considering the current limits on the heavy-light mixings. Our results illustrate the difference between the genuine effects of two nondegenerate heavy Majorana states and the degenerate case where they form a Dirac singlet field. Published by the American Physical Society 2024

Effect of Hopf-Hopf bifurcation on the post-flutter behavior of a three-degree-of-freedom airfoil

The post-flutter dynamics of a three-degree-of-freedom nonlinear airfoil with unsteady aerodynamics are investigated based on the Hopf-Hopf bifurcation theory. Many prior works have relied on Hopf bifurcation theory to predict flutter behavior in airfoil systems. Although this approach facilitates flutter prediction and characterization of post-flutter behavior in numerous scenarios, it may be invalid in specific instances, such as when the Hopf bifurcation of the system degenerates. Therefore, this study focuses on a classical degenerate case of Hopf bifurcation in airfoil systems, specifically the Hopf-Hopf bifurcation. We show that the system undergoes various Hopf-Hopf bifurcations under specific parameter conditions as the center of gravity shifts. The local dynamics near the Hopf-Hopf bifurcation points are represented, including quasiperiodic oscillations on a three-dimensional torus. The airfoil begins to oscillate quasiperiodically after the airflow speed crosses specific Hopf-Hopf bifurcation points. The study also uncovers complex quasiperiodic crises and quasiperiodic hysteresis loops, which have not been reported in previous studies of aeroelastic systems. Then, many singularities and bifurcation curves are obtained near the Hopf-Hopf bifurcation point by semiglobal unfolding. Furthermore, the influence of stall effects on the bifurcation structure of the system is represented. It is shown that the types of Hopf-Hopf bifurcations may vary with the changes of stall effects, influencing the system's semiglobal bifurcation structures consequently. For all Hopf-Hopf bifurcation scenarios, stall effects affect one of the Neimark-Sacker bifurcation curve structures unfolded from the Hopf-Hopf bifurcation point significantly, while the other Neimark-Sacker bifurcation curve experiences minimal impact from stall effects. Moreover, a large nonlinear stall coefficient will postpone the onset of quasiperiodic/chaotic oscillations.

Fine boundary regularity for the singular fractional p-Laplacian

We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded C1,1 domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with s∈(0,1). We prove optimal up-to-the-boundary regularity of u, which is Cs(Ω‾) for any p>1 and, in the singular case p∈(1,2), that u/dΩs has a Hölder continuous extension to the closure of Ω, dΩ(x) meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case p⩾2 is considered.

Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only On Mass Moments

The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found. In this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes. This insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain.

Tensor loop reduction via the Baikov representation and an auxiliary vector

In this paper, we introduce a simple and efficient approach for the general reduction of one-loop integrals. Our method employs the introduction of an auxiliary vector and the identification of the tensor structure as an auxiliary propagator. This key insight allows us to express a wide range of one-loop integrals, encompassing both tensor structures and higher poles, in the Baikov representation. By establishing an integral-by-parts relation, we derive a recursive formula that systematically solves the one-loop reduction problem, even in the presence of various degenerate cases. Our proposed strategy is characterized by its simplicity and effectiveness, offering a significant advancement in the field of one-loop calculations. Published by the American Physical Society 2024

Explicit n-particle harmonic oscillator states in products of SU(1,1) representations

In this paper, we present a simple algorithm for the explicit construction of [Formula: see text]-particle harmonic oscillator states simultaneously belonging to irreducible representations of [Formula: see text] (or [Formula: see text]) and [Formula: see text]. For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with [Formula: see text] and [Formula: see text] are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the [Formula: see text] structure of Bell states.

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

Combined methods for solving degenerate unconstrained optimization problems

UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of ℝ n into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

Necessary and sufficient conditions of entire sub-solutions for a (k1, k2)-type Hessian systems with gradient terms

In this paper, we aim to discuss a class of (k1, k2)-type Hessian system with gradient terms. In the case of k1 = k2 = 1 and 2 ≤ k1, k2 ≤ n, we obtain a sufficient and necessary condition for the existence of the entire admissible sub-solution of the system according to the value range of different parameters, which is also called the generalized Keller–Osserman condition. Based on this, we also discuss the conditions of existence and non-existence of the entire sub-solution, respectively. Finally, we extend the nonlinear terms to the degenerate case and consider the condition of the existence of the positive sub-solution for the above system.

Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map

We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text].

Description of the set of all possible masses at a fixed point of solutions of a truncated matricial Hamburger moment problem

We describe the set of all possible masses at a fixed point of the real axis in the context of the most general case of a truncated matricial (possibly degenerate) Hamburger moment problem. This set is a matricial interval, which we explicitly construct from the given data in the non-degenerate and degenerate cases simultaneously. In particular, we determine the maximum possible mass.

Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation

The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general M−SM-(N−M)-soliton solutions (0≤M≤N), including M SMs and N−M solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., N-SM solutions (M=N) and N-soliton solutions (M=0). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.

Cooperative effects in dense cold atomic gases including magnetic dipole interactions

We theoretically investigate cooperative effects in cold atomic gases exhibiting both electric and magnetic dipole-dipole interactions, such as occurs, for example, in clouds of dysprosium atoms. After introducing a general framework capturing both the quantum degenerate and nondenegerate cases, we focus on the emergence of tailorable spin models in the quantum nondegenerate regime. In the low-excitation limit, we provide analytical and numerical results detailing the effect of magnetic interactions on the directionality of scattered light and characterize sub and superradiant effects. Published by the American Physical Society 2024

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.