Symmetry Reduction Research Articles

Some lower dimensional quantum field theories reduced from Chern-Simons gauge theories

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on S3 and the reduced quantum field theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding a topological theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function. Published by the American Physical Society 2024

Scaling symmetry reductions of coupled KdV systems

Abstract In this paper we discuss the Painlevé reductions of coupled KdV systems. We start by comparing the procedure with that of stationary reductions. Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the t2 flow of the KdV and DWW hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical structure to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case.

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

Dislocation‐induced local symmetry reduction in single‐crystal KNbO3 observed by Raman spectroscopy

AbstractIn recent years, dislocation‐tuned functionalized ceramics have become one of the focal points of modern materials research due to their outstanding properties. These range from improved electrical conductivity and ferroelectric properties to dislocation‐enhanced toughening and superconductivity, caused by local strain fields. The aim of this work is to investigate the dislocation‐induced structural changes in single‐crystal KNbO3 by micro‐Raman spectroscopy. Dislocation‐rich regions with tailored densities have been generated using the Brinell indenter scratching method. The influence of the dislocations on the single‐crystal structure can be observed in the Raman spectra. The activation of additional Raman modes is observed, which does not occur in regions of low dislocation density. This observation is confirmed by DFT calculations of vibrational modes and is attributed to a reduction in the crystal symmetry due to increased defect densities in plastically deformed KNbO3. In addition, an increase in compressive stress at higher dislocation densities can be demonstrated by a blueshift in Raman mode positions.

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

A density functional theory study of nitrogen vacancy center in lonsdaleite

Abstract Lonsdaleite is a carbon allotrope and metastable form of diamond that demonstrates superior mechanical properties over cubic diamond. Here, we report the results of density functional theory (DFT) and molecular dynamics (MD) study of neutral and negative nitrogen vacancy center in lonsdaleite. Interestingly, neutral (NV0) and negative (NV-1) nitrogen-vacancy center in lonsdaleite display a remarkable splitting in the two degenerate ex and ey excited states nearly around ~ 0.5 eV for NV0 and 0.2 eV for NV-1. The thermal stability, dynamic stability, band structure, density of states, and optical properties are computed. The DFT and MD calculations reveal that geometrical structure of NV center in lonsdaleite is both thermally and dynamically stable. In addition, the findings show that NV0 and NV-1centers in lonsdaleite demonstrate splitting in the zero-phonon line (ZPL) due to symmetry reduction from C3v to C1h with respect to NV center in cubic diamond. Furthermore, the results indicate that ZPL falls around ~ 1.76 and 2.25 eV for NV0, while it lies around 1.91 and 2.19 eV for NV-1.

Deterministic and stochastic geometric mechanics for Hall magnetohydrodynamics

We derive new models of stochastic Hall magnetohydrodynamics (MHD) by applying Lie group symmetry reduction in a stochastic Euler–Poincaré variational principle. The resulting new theory of stochastic Hall MHD has potential applications for uncertainty quantification and data assimilation in space plasma (space weather), planetary science and solar physics. The stochastic geometric mechanics approach we take here produces coordinate-free results that may then be applied in a variety of spatial configurations.

Analysis of Lie symmetry, bifurcations with phase portraits, sensitivity and diverse W − M-shape soliton solutions for the (2 + 1)-dimensional evolution equation

This research investigates the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation, a key model in nonlinear wave dynamics, particularly for shallow water waves. By integrating the features of the KP and BBM equations, it offers a comprehensive framework for studying the propagation, stability, and interactions of long waves. Using Lie group analysis for symmetry reductions and bifurcation with phase portraits to explore dynamic behaviour, the study also applies chaos theory to examine system features. Additionally, the new extended direct algebraic method (NEDAM) is employed to derive novel soliton solutions, including dark, bright-dark, W-shape, singular, periodic, and mixed forms. Sensitivity analysis is performed, and 3D, 2D, contour, and density plots visually demonstrate the results. These findings show that NEDAM effectively extracts exact solitons, providing a basis for further studies in nonlinear wave theory and its applications in engineering and physics.

Residual symmetry and interactions between solitons and periodic waves of the Drinfeld-Sokolov-Satsuma-Hirota system

Abstract In this paper, the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system is studied by using residual symmetry and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to a Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system is proved to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

Residual Symmetry and Interaction Solutions of the (2+1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

In this paper, we investigate the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (gCBS) equation by using the residual symmetry analysis and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the gCBS equation is localized into a Lie point symmetry in a prolonged system and a new Bäcklund transformation of this equation is obtained. By applying the standard Lie symmetry method to the prolonged gCBS system, new symmetry reduction solutions of the gCBS equation are obtained. The gCBS equation is proved to be CRE integrable and new Bäcklund transformations of it are obtained, from which interaction solutions between solitons and periodic waves are generated and analyzed.

Solving clustered low-rank semidefinite programs arising from polynomial optimization

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

Partially invariant solution with an arbitrary surface of blow-up for the gas dynamics equations admitting pressure translation

We applied a method of symmetry reduction to the gas dynamics equations with a special form of the equation of state. This equation of state is a pressure represented as the sum of a density and an entropy functions. The symmetry Lie algebra of the system is 12-dimensional. One, two and three-dimensional subalgebras were considered. In this article, four-dimensional subalgebras are considered for the first time. Specifically, invariants are calculated for 50 four-dimensional subalgebras. Using invariants of one of the subalgebras, a symmetry reduction of the original system is calculated. The reduced system is a partially invariant submodel because one gas-dynamic function cannot be expressed in terms of the invariants. The submodel leads to two families of exact solutions, one of which describes the isochoric motion of the media, and the other solution specifies an arbitrary blow-up surface. For the first family of solutions, the particle trajectories are parabolas or rays; for the second family of solutions, the particles move along cubic parabolas or straight lines. From each point of the blow-up surface, particles fly out at different speeds and end up on a straight line at any other fixed moment in time. A description of the motion of particles for each family of solutions is given.

Lie symmetry reductions, exact solutions and soliton dynamics to Burgers equation

Lie symmetry reductions, exact solutions and soliton dynamics to Burgers equation

Dynamics of quasi-periodic, bifurcation, sensitivity and three-wave solutions for (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation.

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models.

Structural origin of spin-splitting anisotropy in janus dichalcogenides monolayers under pressure

Abstract Janus transition-metal dichalcogenides (TMDs) have drawn a great deal of attention because of their mirror plane symmetry breaking that allows the emergence of a built-in out-of-plane dipole which determine superior piezoelectric and spin-related properties. Furthermore, it has been shown in the recent literature that pressure application is capable of modulating spin-related phenomena in this class of materials. Generally, the spin-splitting presence in real systems is explored in terms of point group symmetry reduction using solely group theory arguments. However, we seek to associate the enhancement of spin-splitting in Janus TMDs monolayers by searching the most important local asymmetries responsible for the symmetry lowering that leads the monolayer larger spin-splitting energies. In this sense, we seek to unveil a possible structural descriptor that correlates with subbands splitting magnitude in Janus TMDs. To accomplish this, we performed a detailed first-principles investigation into WSSe Janus monolayers under biaxial in-plane strain to find that pressure induces a symmetry lowering from the C 3v to the C s point group. From these observations, we found that in-plane angle asymmetries between the chalcogens yield a distortion metric that can serve as a descriptor for enhanced spin-splitting in Janus WSSe since it strongly correlates with spin-splitting energies. Hence, our work establishes that, rather than solely global symmetry analysis, specific local distortions provide a key design principle to achieve strong spin-splitting in 2D Janus TMDs.

Perpendicular magnetic anisotropy in multilayers arising from the interplay of thermal strains and diffusion-driven plastic deformation

Magnetic films with perpendicular magnetic anisotropy (PMA) are the basis for efficient memory-storage and future spintronic devices. PMA predominantly stems from surface effects, e.g., symmetry reduction, at the magnetic-layer interface and quickly decays with increasing layer thickness. Strong PMA is thus typically observed in sub-nanometer multilayers with alternating magnetic and noble metals. Using in-situ temperature-dependent measurements of ferromagnetic resonance it is demonstrated that strong PMA can be achieved in Si/Ni multilayers with individual layer as thick as 16 nm. Here, PMA is imposed by thermal strains acting via magnetoelastic coupling and directly depends on the annealing temperature and time, which regulate the diffusion-driven plastic deformation. This opens a way to PMA-film fabrication that avoids complex and resource-intensive production of atomically thin multilayers.

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

Invariant analysis, invariant subspace method and conservation laws of the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system

In this paper, we investigate the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system (MFBKKS) with Riemann–Liouville time fractional derivative and integer order y-derivative. This system models dispersive and nonlinear long gravity waves propagating in shallow water in two horizontal directions with time memories. Lie symmetry analysis and invariant subspace method are distinctly employed to construct the exact solutions to the MFBKKS. Initially, the Lie algebras admitted by MFBKKS are obtained with the help of Lie symmetry analysis. Then, we establish the commutative table, adjoint relations and adjoint transformation matrix. Specifically, the one-dimensional optimal system is established correspondingly, and symmetry reduction is performed. In particular, (2+1)-dimensional MFBKKS is reduced to (1+1)-dimensional mixed fractional system using Erdélyi–Kober fractional differential operator. Further, the power series solution of MFBKKS is constructed via power series method. By applying invariant subspace method, we obtain more exact solutions of MFBKKS. In addition, the conservation laws are derived by new Noether theorem. Finally, the three-dimensional diagrams of some obtained solutions are demonstrated utilizing Matlab for visualization, and some of the calculations were verified using computational packages Maple for symbolic computation.

Lie group analysis, solitary wave solutions and conservation laws of Schamel Burger’s equation

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.