One-dimensional Symmetry Research Articles

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

Spreading, flattening, and logarithmic lag for reaction-diffusion equations in $\mathbb{R}^{N}$: Old and new results

This paper is concerned with the large-time dynamics of bounded solutions of reaction-diffusion equations with bounded or unbounded initial support in \R^{N} . We start with a survey of some old and recent results on the spreading speeds of the solutions and their asymptotic local one-dimensional symmetry. We then derive some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. Lastly, we reclaim some known results about the logarithmic lag between the position of the solutions and that of planar or spherical fronts expanding with minimal speed, for almost-planar or compactly supported initial conditions. We then prove some new logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. These estimates entail in particular that the same lag as for compactly supported initial data holds true for a class of unbounded initial supports. The paper also contains some related conjectures and open problems.

Universal Entanglement Spectrum in One-Dimensional Gapless Symmetry Protected Topological States.

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.

Sliding method and one-dimensional symmetry for p-Laplace equations

Sliding method and one-dimensional symmetry for p-Laplace equations

Enhancing particle string detection in electrorheological plasmas using asymmetrical kernel convolutional networks

Under different plasma conditions and electric fields in a complex plasma the plasma particles organize themselves in a string-like or chain-like manner. A phase transition from string-like to an isotropic particle distribution is observed at different electrical conditions. The streaming of charged ions around plasma particles with the surrounding electric field gives the plasma its electrorheological properties. The visibility of individual particles in a complex plasma opens up the opportunity to examine properties and phase transitions of such electrorheological fluids in detail. Because of the limited one-dimensional symmetry, determining the configuration of a particle and recognizing strings in particle distributions is not always straightforward. Several approaches have already been used to analyse particle clouds while either considering each particle locally or considering the particle cloud as a whole without providing information about single particle configurations. This paper presents a new machine learning approach that takes advantage of particle distributions over the entire particle cloud and detects all string-like particles at once, using a convolutional neural network in form of an encoder-decoder network with asymmetric kernel convolutions. This not only enhances the result quality but also accelerates the evaluation process, possibly enabling real-time analyses on electrorheological phase transitions, while achieving an accuracy of over 95% on manually labelled data.

A Liouville-type theorem and one-dimensional symmetry of solutions for elliptic equations with general gradient nonlinearity

In this paper, we are concerned with the following elliptic equation involving a general gradient nonlinearity:−Δv=μ|v|p−1v+G(∇v)inΩ, where n≥1, either Ω=Rn is the whole space or Ω=R+n={x=(x1,…,xn)∈Rn:xn>0} is the half space, μ≥0, p∈R and G:Rn→R satisfies some suitable conditions. One particular case of G is G(γ)=∑i=1kai|γ|pi with ai>0 and pi>1 for any 1≤i≤k. When Ω=Rn, we establish a Bernstein estimate to prove some Liouville-type results of the equation under certain conditions on μ and p. When Ω=R+n, by combining the Bernstein estimates with a moving planes argument, we obtain the one-dimensional symmetry of the solution v with v=0 on ∂R+n. We note that the exponent p is allowed to be negative in our results. Finally, we obtain a gradient estimate in some special cases with μ>0 and p>1.

Monotonicity and one-dimensional symmetry of solutions for the weighted fractional parabolic equations on the whole space

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions.

Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

Abstract We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ , \left\{\begin{aligned} \displaystyle{}\frac{\partial u}{\partial t}(x,t)+(-% \Delta)_{p}^{s}u(x,t)&\displaystyle=f(t,u(x,t)),&\hskip 10.0pt(x,t)&% \displaystyle\in\Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle>0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle=0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega^{c}\times\mathbb{R},\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , p ≥ 2 {p\geq 2} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian, f ⁢ ( t , u ) {f(t,u)} is some continuous function, the domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is unbounded and Ω c = ℝ n ∖ Ω {\Omega^{c}=\mathbb{R}^{n}\setminus\Omega} . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ {t\in\mathbb{R}} . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.

Sensitivity Study on Typical Parameters of Underwater Explosion Numerical Simulation

In the numerical simulation research for underwater explosion, the selection of simulation parameters has a great influence on the results of numerical calculation. Based on the one-dimensional spherical symmetry model, this paper systematically studies the influence of three factors: grid size, water state equation and artificial viscosity coefficient on the important physical parameters of water explosion when TNT explosive is exploded in water. The important physical parameters selected for the explosion in water are the shock wave intensity, the maximu m radius of the bubble and the pulsation period of the bubble. A series of studies are carried out on underwater explosions with different grid sizes, so as to obtain the corresponding recommended grids that meet the calculation accuracy. The influence of different water state equations on the simulation results is discussed. The effect of artificial viscosity coefficient on the simulation results is analyzed. Finally, the similarity law of the model is studied to verify the universality of the model parameters.

Rigidity of phase transitions for the fractional elliptic Gross-Pitaevskii system

We prove the monotonicity and one-dimensional symmetry of solutions to non-cooperative fractional elliptic systems with uniform limits. In particular, our result holds for phase transitions for the fractional elliptic Gross-Pitaevskii system. The main tool we use is a sliding method adapted to fractional elliptic systems.

Lie point symmetry infinitesimals, optimal system, power series solution, and modulational gain spectrum to the mathematical Noyes–Field model of nonlinear homogeneous oscillatory Belousov–Zhabotinsky reaction

Introduction:The chemical oscillators are identified as open system that demonstrate periodic changes in the concentration of some reaction species as a result of intricate physico-chemical mechanisms which can lead to bi-stability, the occurrence of limit cycle attractors, the emergence of spiral waves and turing patterns, and finally, deterministic chaos. Objectives:The main objective of this paper is to analyze the simple Noyes–Field governing system of differential equations for the nonlinear Belousov–Zhabotinsky reaction which delineates the non-linear oscillatory behavior of chemical systems that occurs in the homogeneous media. Methodology:The Lie symmetry invariance analysis performed to extract the symmetries infinitesimal generators and the adjoint representation carried out to develop optimal system for the obtained Lie vectors. The significant power series approach applied to obtain the analytical solution. The modulation instability criteria ensured the stability of nonlinear oscillatory Belousov–Zhabotinsky reaction process. Results:The one-dimensional Lie symmetry generators algebra of the mathematical Noyes–Field governing system for oscillatory reaction is established. Furthermore, similarity reductions are carried out as well as the development of an optimal system of the sub-algebras. The similarity transformation technique converted the controlling system to ordinary differential equations and generates the large quantity of analytical traveling wave solutions. Moreover, the closed-form analytical solution for the proposed homogeneous nonlinear oscillatory chemical process is secured. The (MI) gain spectrum graphically visualized with the suitable choice of arbitrary parameters. Conclusion:The graphical performance of the Noyes–Field model solution at various settings reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

One-dimensional symmetry of solutions to non-cooperative elliptic systems

We prove the monotonicity and one-dimensional symmetry of solutions to non-cooperative elliptic systems in RN with uniform limits under mild assumptions on the nonlinearities. These problems arise in the study of domain walls and interface layers of two-component Bose–Einstein condensates in the segregation regime. We also derive analogous results for systems in half-spaces and strips. We introduce a variant of the sliding method for non-cooperative elliptic systems to prove our results.

Spreading sets and one-dimensional symmetry for reaction-diffusion equations

Spreading sets and one-dimensional symmetry for reaction-diffusion equations

One-dimensional symmetry of positive bounded solutions to the subcubic and cubic nonlinear Schrödinger equation in the half-space in dimensions $$N=4,5$$

We are concerned with the half-space Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta v+v=|v|^{p-1}v &{} \text {in}\ {\mathbb {R}}^N_+, \\ v=c\ \text {on}\ \partial {\mathbb {R}}^N_+, &{}\lim _{x_N\rightarrow \infty }v(x',x_N)=0\ \text {uniformly in}\ x'\in {\mathbb {R}}^{N-1}, \end{array}\right. \end{aligned}$$where \({\mathbb {R}}^N_+=\{x\in {\mathbb {R}}^N \ : \ x_N>0\}\) for some \(N\ge 2\), and \(p>1\), \(c>0\) are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number \(c_p\in (1,\sqrt{e})\), depending only on p, such that for \(0<c<c_p\) there are infinitely many bounded positive solutions, whereas, for \(c>c_p\) there are no bounded positive solutions. They also posed as an interesting open question whether the one-dimensional solution is the unique bounded positive solution in the case where \(c = c_p\). If \(N=2, 3\), we recently showed this one-dimensional symmetry property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the proof of De Giorgi’s conjecture in low dimensions. Here, we first focus on the case \(1<p<3\) and prove this uniqueness property in dimensions \(2\le N\le 5\). Then, for the cubic NLS, where \(p=3\), we establish this for \(2\le N \le 4\). Our approach is completely different and relies on showing that a suitable auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in \({\mathbb {R}}^{N-1}\), for which an optimal Liouville type result is available.

Effective field theory of fluctuating wall in open systems: from a kink in Josephson junction to general domain wall

We investigate macroscopic behaviors of fluctuating domain walls in nonequilibrium open systems with the help of the effective field theory based on symmetry. Since the domain wall in open systems breaks the translational symmetry, there appears a gapless excitation identified as the Nambu-Goldstone (NG) mode, which shows the non-propagating diffusive behavior in contrast to those in closed systems. After demonstrating the presence of the diffusive NG mode in the (2+1)-dimensional dissipative Josephson junction, we provide a symmetry-based general analysis for open systems breaking the one-dimensional translational symmetry. A general effective Lagrangian is constructed based on the Schwinger-Keldysh formalism, which supports the presence of the gapless diffusion mode in the fluctuation spectrum in the thin wall regime. Besides, we also identify a term peculiar to the open system, which possibly leads to the instability in the thick-wall regime or the nonlinear Kardar-Parisi-Zhang coupling in the thin-wall regime although it is absent in the Josephson junction.

Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a C1,α estimates for the “interfaces” (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole RN, for N≤4, answering positively a conjecture of Fernández-Real and Ros-Oton.Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.

A note on one-dimensional symmetry for Hamilton–Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative \(L^q\)-viscosity solutions of the equation $$\begin{aligned} -\mathcal {M}_{\lambda , \Lambda }^{\pm }(D^2u)\pm |Du|^p=0, x\in \mathbb {R}_+^n, \end{aligned}$$with boundary condition \(u(\tilde{x},0)=M\ge 0, \tilde{x}\in \mathbb {R}^{n-1}\), where \(\mathcal {M}_{\lambda , \Lambda }^{\pm }\) are the Pucci’s operators with parameters \(\lambda , \Lambda \in \mathbb {R}_+\) \(0<\lambda \le \Lambda \) and \(p>1\). The results are an extension of the results by Porretta and Véron (Adv Nonlinear Stud 6:351–378, 2006) for the case \(p\in (1,2]\) and by Filippucci et al. (Commun Partial Differ Equ 45(2):321–349, 2019) for the case \(p>2\), both for the Laplacian case (i.e. \(\lambda =\Lambda =1\)). As an application in the case \(p>2\), we prove a sharp Bernstein estimation for \(L^q\)-viscosity solutions of the fully nonlinear equation $$\begin{aligned} -\mathcal {M}_{\lambda , \Lambda }^{\pm }(D^2u)= |Du|^p+f(x), \quad x\in \Omega , \end{aligned}$$with boundary condition \(u=0\) on \(\partial \Omega \), where \(\Omega \subset \mathbb {R}^n\).

Linear projections of joint symmetry and independence applied to exact testing treatment effects based on multidimensional outcomes

The growing need for analyzing multivariate aspects of joint data distributions is reinforced by a diversity of experiments based on dependent outcomes. In this sense, different contexts of joint symmetry of data distributions have been dealt with extensively in both theory and practice. Univariate characterizations of properties of multivariate distributions can allow the reduction of the original problem to a substantially simpler one. We focus on research scenarios when vectors x and Ax are identically distributed, where A is a diagonal matrix and absolute values of A’s elements equal to one. It is shown that these scenarios are attractive in new characterizations of joint or mutual independence between random variables. We establish projections of the joint symmetry and independence via the one-dimensional symmetry of linear combinations of x’s components and their interactions. These projections are the most revealing of the multivariate data distribution. The usefulness of the linear projections is exemplified by constructing an efficient nonparametric exact test for joint treatment effects. In this framework, an algorithm for implementing linear projection-based tests is proven. Numerical studies based on generated vectors and a real dataset show that the proposed test can exhibit high and stable power characteristics. The present method can be also used for testing independence between symmetric random vectors.

Thermal decay of magnons in MnF2

It is shown that for nearly all magnets only two types of magnon dispersions occur. For the first group of magnets, magnon dispersions are as for the linear spin chain (a sine-function of wave-vector for antiferromagnets but a sine-function squared for ferromagnets), quite independent of the lattice structure. Since the observed magnon dispersions are specific to the individual domain, the identical magnon dispersions show that the domains of all magnets are very similar in that all spins are perfectly aligned along the domain axis. The linear chain dispersion is explained by a strong coupling of the spins to the one-dimensionally ordered boson field in each domain. The bosons were identified as essentially magnetic dipole radiation generated essentially through stimulated emission by the precessing spins. In the ordered state of the boson field, the bosons are in a coherent and one-dimensional state, like the radiation field of a LASER. For the second group of magnets, magnon dispersions exhibit an analytical crossover from a power function of wave vector (∼qx) plus a gap to the dispersion of the linear spin chain. Occasionally it turns out to be necessary to add a phase shift in the argument of the sine function. The functionality of the local exchange interactions is restricted to determine the relative orientation of neighboring spins, i.e. to decide between the sine-function and the sine-function squared and to define the absolute values of the magnon energies. As a consequence, instead by exchange interactions, magnon dispersions have to be characterized by four parameters: the gap, the pre-factor of the ∼qx power function, the amplitude of the sine-function and the phase-shift in the argument of the sine function. For MnF2 the temperature dependence of these four parameters is evaluated, as far as experimental data are available. The exponent x is the only parameter that is characteristic of the dynamic dimensionality of the system and is temperature-independent. For MnF2 with a half-integer spin of S = 5/2 and a one-dimensional dynamic symmetry x = 3/2.

One-dimensional symmetry of positive bounded solutions to the nonlinear Schrödinger equation in the half-space in two and three dimensions

One-dimensional symmetry of positive bounded solutions to the nonlinear Schrödinger equation in the half-space in two and three dimensions