In this paper,the(1+1)-dimensional classical Boussinesq-Burgers(CBB)system is extended to a(4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm.The Lax integrability,symmetry integ...In this paper,the(1+1)-dimensional classical Boussinesq-Burgers(CBB)system is extended to a(4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm.The Lax integrability,symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given.Meanwhile,for illustration,an exact solution of a(1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.展开更多
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical...The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.展开更多
We show that higher-dimensional integrable systems including the (2+1)-dimensional generalized sine-Gordon equation and the (2+1)-dimensional complex mKdV equation are associated with motions of surfaces inducedby end...We show that higher-dimensional integrable systems including the (2+1)-dimensional generalized sine-Gordon equation and the (2+1)-dimensional complex mKdV equation are associated with motions of surfaces inducedby endowing with an extra space variable to the motions of curves on S^2(R) and S^3(R).展开更多
Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-- Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coetticient ...Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-- Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coetticient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty prlnciple and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is positive. This result is different from the known result at present. Our method is valid not only for four-dimensional spacetimes but also for higher-dimensional spacetimes. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the entropy correction of the complicated spacetime.展开更多
By means of the standard truncated Painlevé expansion and a special B?cklund transformation, the higher-dimensional coupled Burgers system (HDCB) is reduced to a linear equation, and an exact multisoliton excitat...By means of the standard truncated Painlevé expansion and a special B?cklund transformation, the higher-dimensional coupled Burgers system (HDCB) is reduced to a linear equation, and an exact multisoliton excitation is derived. The evolution properties of the multisoliton excitation are investigated and some novel features or interesting behaviors are revealed. The results show that after interactions for dromion-dromion, solitoff-solitoff, and solitoff-dromion, they are combined with some new types of localized structures, which are similar to classic particles with completely nonelastic behaviors.展开更多
Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,wi...Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.展开更多
In this paper,we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem.Based on the existence,an asymptotical analysis of a steplike contrast ...In this paper,we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem.Based on the existence,an asymptotical analysis of a steplike contrast structure (i.e.,an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection.In the framework of this paper,we propose a first integral condition,under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space.Then,the step-like contrast structure is constructed,and the internal transition time is determined.Meanwhile,the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained.Finally,an example is presented to illustrate the result.展开更多
Based on the theory of Klein-Gordon scalar field particles, the Hawking radiation of a higher- dimensional Kerr-anti-de Sitter black hole with one rotational parameter is investigated using the beyond semi-classical a...Based on the theory of Klein-Gordon scalar field particles, the Hawking radiation of a higher- dimensional Kerr-anti-de Sitter black hole with one rotational parameter is investigated using the beyond semi-classical approximation method. The corrections of quantum tunnelling probability, Hawking temperature and Bekenstein-Hawking entropy are also included.展开更多
Based on a theory of extra dimensional confinement of quantum particles [E. R. Hedin, Physics Essays, 2012, 25（2）： 177], a simple model of a nucleon nucleon （NN） central potential is derived which quantitatively ...Based on a theory of extra dimensional confinement of quantum particles [E. R. Hedin, Physics Essays, 2012, 25（2）： 177], a simple model of a nucleon nucleon （NN） central potential is derived which quantitatively reproduces tile radial profile of other models, without adjusting any free pa- rameters. It is postulated that a higher-dimensional simple harmonic oscillator confining potential localizes particles into three-dimensional （3D） space, but allows for an ewmescent penetration of the particles into two higtmr spatial dimensions. Producing an effect identical with the relativistic quan- tum phenolnenon of zitterbewegung, the higher-dimensional oscillations of amplitude h（mc） call be alternatively viewed as a localized curvature of 3D space back and forth into the higher dimensions. The overall spatial curvature is proportional to the particle＇s extra-dimensional ground state wave function in tile higher-dimensional harmonic confining potential well. Minimizing the overlapping curvature （proportional to the energy） of two particles in proximity to each other, subject to the constraint that for the two particles to occupy the same spatial location one of them must be excited into the 1st excited state of the harmonic potential well, gives the desired NN potential. Specifying only the imcleon masses, the resulting potential well and repulsive core reproduces the radial profile of several published NN central potential models. In addition, the predicted height of the repulsive core, when used to estimate the maximum neutron star mass, matches well with the best estimates from relativistic theory incorporating standard nuclear matter equations of state. Nucleon spin, Coulomb interactions, and internal nucleon structure are not considered in the theory as presented in this article.展开更多
We apply the WKB approximation method,matrix method,and finite difference method to study the gravitational quasi-normal modes of charged spherically symmetric black holes surrounded by quintessence fluid in Rastall g...We apply the WKB approximation method,matrix method,and finite difference method to study the gravitational quasi-normal modes of charged spherically symmetric black holes surrounded by quintessence fluid in Rastall gravity.By comparing the spherically symmetric spacetime metric of charged black holes surrounded by quintessence fluid in Rastall gravity with that of general relativity,we can find that the modifications to general relativity in this modified gravity theory can be described by parameters such asλ,Q,and C_(a),etc.In four-dimensional spacetime,we investigate the impact of charge Q and parameter C_(a) on the gravitational quasi-normal modes of charged black holes surrounded by quintessence field in Rastall gravity.The aim is to search for observational evidence of such black holes in astrophysical observations and,consequently,test the validity of Rastall theory.In five-dimensional(5D)spacetime,we study the impact of the parameter C_(a) on the gravitational quasi-normal modes of Rastall black holes surrounded by quintessence field and summarize the corresponding variation patterns.展开更多
The generalized uncertainty relation is introduced to calculate quantum statistic entropy of a black hole. By using the new equation of state density motivated by the generalized uncertainty relation, we discuss entro...The generalized uncertainty relation is introduced to calculate quantum statistic entropy of a black hole. By using the new equation of state density motivated by the generalized uncertainty relation, we discuss entropies of Bose field and Fermi field on the background of the five-dimensional spacetime. In our calculation, we need not introduce cutoff. There is not the divergent logarithmic term as in the original brick-wall method. And it is obtained that the quantum statistic entropy corresponding to black hole horizon is proportional to the area of the horizon. Fhrther it is shown that the entropy of black hole is the entropy of quantum state on the surface of horizon. The black hole＇s entropy is the intrinsic property of the black hole. The entropy is a quantum effect. It makes people further understand the quantum statistic entropy.展开更多
The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional...The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional derivative concept,while the model is solved via the full-spectral method(FSM)and the semi-spectral scheme(SSS).The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques.The SSS is developed by discretizing the time variable,and the space domain is collocated by using equal points.A detailed comparative analysis is made through graphs for various parameters and tables with existing literature.The contour graphs are made to show the behaviors of the velocity and magnetic fields.The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows,and the concept may be extended for variable order models arising in MHD flows.展开更多
We discuss an object from algebraic topology,Hopf invariant,and reinterpret it in terms of the φ-mappingtopological current theory.The main purpose of this paper is to present a new theoretical framework,which can di...We discuss an object from algebraic topology,Hopf invariant,and reinterpret it in terms of the φ-mappingtopological current theory.The main purpose of this paper is to present a new theoretical framework,which can directlygive the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclideanspace R^(2n-1).For the sake of this purpose we introduce a topological tensor current,which can naturally deduce the(n-1)-dimensional topological defect in R^(2n-1) space.If these (n-1)-dimensional topological defects are closed orientedsubmanifolds of R^(2n-1),they are just the (n-1)-dimensional knots.The linking number of these knots is well defined.Using the inner structure of the topological tensor current,the relationship between Hopf invariant and the linkingnumbers of the higher-dimensional knots can be constructed.展开更多
ⅠIn the recent years, with the development of the superstring theory (requiring (1+9) dimensional space time) and the application of the Kaluza-Klein theory to the research of the very early phases of the universe (r...ⅠIn the recent years, with the development of the superstring theory (requiring (1+9) dimensional space time) and the application of the Kaluza-Klein theory to the research of the very early phases of the universe (requiring (1+10) dimensional space-time), higher-dimensional physics has assumed a high measure of importance. Emelyanov et al. have in detail evaluated the study situation on the higher-dimensional space-time.展开更多
We consider both gauged and ungauged minimal supergravities in five dimensions and analyse the charged rotating solutions with two equal angular momenta J.When the electric charge Q∼J^(2/3) with some specific coeffic...We consider both gauged and ungauged minimal supergravities in five dimensions and analyse the charged rotating solutions with two equal angular momenta J.When the electric charge Q∼J^(2/3) with some specific coefficient,we find new extremal black objects emerge that are asymptotic to either Minkowski or global AdS spacetimes and can be best described as degenerate black rings.Their near-horizon geometry is locally AdS3×S^(2),where the periodic U(1)fibre coordinate in S 3 untwists and collapses to be the degenerate part of the AdS3 horizon.It turns out that there are two branches of extremal rotating black holes,starting as the extremal RN black holes of the same mass,but opposite charges.With the increasing of the angular momentum,they will join to become the same degenerate black ring,where the Gibbs free energies however are not continuous at the joining.For the same Q(J)relation,we find that there is in addition a rotating soliton whose mass is smaller than that of the degenerate black ring.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11871396,12271433).
文摘In this paper,the(1+1)-dimensional classical Boussinesq-Burgers(CBB)system is extended to a(4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm.The Lax integrability,symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given.Meanwhile,for illustration,an exact solution of a(1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.
基金sponsored by the National Natural Science Foundations of China(Nos.12235007,11975131,11435005,12275144,11975204)KC Wong Magna Fund in Ningbo UniversityNatural Science Foundation of Zhejiang Province No.LQ20A010009。
文摘The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.
基金National Natural Science Foundation of China under Grant No.10671156the Program for New Century Excellent Talents in Universities under Grant No.NCET-04-0968
文摘We show that higher-dimensional integrable systems including the (2+1)-dimensional generalized sine-Gordon equation and the (2+1)-dimensional complex mKdV equation are associated with motions of surfaces inducedby endowing with an extra space variable to the motions of curves on S^2(R) and S^3(R).
基金The project supported by the Natural Science Foundation of Shanxi Province under Grant No. 2006011012 tCorresponding author,
文摘Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-- Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coetticient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty prlnciple and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is positive. This result is different from the known result at present. Our method is valid not only for four-dimensional spacetimes but also for higher-dimensional spacetimes. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the entropy correction of the complicated spacetime.
文摘By means of the standard truncated Painlevé expansion and a special B?cklund transformation, the higher-dimensional coupled Burgers system (HDCB) is reduced to a linear equation, and an exact multisoliton excitation is derived. The evolution properties of the multisoliton excitation are investigated and some novel features or interesting behaviors are revealed. The results show that after interactions for dromion-dromion, solitoff-solitoff, and solitoff-dromion, they are combined with some new types of localized structures, which are similar to classic particles with completely nonelastic behaviors.
基金the North-West University,Mafikeng campus for its continued support.
文摘Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.
基金supported by National Natural Science Foundation of China (Grant Nos. 11071075,11171113)the NSFC-the Knowledge Innovation Program of the Chinese Academy of Science(Grant Nos. 30921064,90820307)+1 种基金Shanghai Municipal Natural Science Foundation (Grant Nos. 10ZR1409200,11ZR1410200)E-Institutes of the Shanghai Municipal Education Commission (Grant No. N.E03004)
文摘In this paper,we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem.Based on the existence,an asymptotical analysis of a steplike contrast structure (i.e.,an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection.In the framework of this paper,we propose a first integral condition,under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space.Then,the step-like contrast structure is constructed,and the internal transition time is determined.Meanwhile,the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained.Finally,an example is presented to illustrate the result.
基金Supported by National Natural Science Foundation of China (10778719)Natural Science Foundation of Hainan Province(109004)
文摘Based on the theory of Klein-Gordon scalar field particles, the Hawking radiation of a higher- dimensional Kerr-anti-de Sitter black hole with one rotational parameter is investigated using the beyond semi-classical approximation method. The corrections of quantum tunnelling probability, Hawking temperature and Bekenstein-Hawking entropy are also included.
文摘Based on a theory of extra dimensional confinement of quantum particles [E. R. Hedin, Physics Essays, 2012, 25（2）： 177], a simple model of a nucleon nucleon （NN） central potential is derived which quantitatively reproduces tile radial profile of other models, without adjusting any free pa- rameters. It is postulated that a higher-dimensional simple harmonic oscillator confining potential localizes particles into three-dimensional （3D） space, but allows for an ewmescent penetration of the particles into two higtmr spatial dimensions. Producing an effect identical with the relativistic quan- tum phenolnenon of zitterbewegung, the higher-dimensional oscillations of amplitude h（mc） call be alternatively viewed as a localized curvature of 3D space back and forth into the higher dimensions. The overall spatial curvature is proportional to the particle＇s extra-dimensional ground state wave function in tile higher-dimensional harmonic confining potential well. Minimizing the overlapping curvature （proportional to the energy） of two particles in proximity to each other, subject to the constraint that for the two particles to occupy the same spatial location one of them must be excited into the 1st excited state of the harmonic potential well, gives the desired NN potential. Specifying only the imcleon masses, the resulting potential well and repulsive core reproduces the radial profile of several published NN central potential models. In addition, the predicted height of the repulsive core, when used to estimate the maximum neutron star mass, matches well with the best estimates from relativistic theory incorporating standard nuclear matter equations of state. Nucleon spin, Coulomb interactions, and internal nucleon structure are not considered in the theory as presented in this article.
基金Project supported by the National Natural Science Foundation of China(Grant No.42230207)the Fundamental Research Funds for the Central Universities,China University of Geosciences(Wuhan)(Grant No.G1323523064)。
文摘We apply the WKB approximation method,matrix method,and finite difference method to study the gravitational quasi-normal modes of charged spherically symmetric black holes surrounded by quintessence fluid in Rastall gravity.By comparing the spherically symmetric spacetime metric of charged black holes surrounded by quintessence fluid in Rastall gravity with that of general relativity,we can find that the modifications to general relativity in this modified gravity theory can be described by parameters such asλ,Q,and C_(a),etc.In four-dimensional spacetime,we investigate the impact of charge Q and parameter C_(a) on the gravitational quasi-normal modes of charged black holes surrounded by quintessence field in Rastall gravity.The aim is to search for observational evidence of such black holes in astrophysical observations and,consequently,test the validity of Rastall theory.In five-dimensional(5D)spacetime,we study the impact of the parameter C_(a) on the gravitational quasi-normal modes of Rastall black holes surrounded by quintessence field and summarize the corresponding variation patterns.
基金The project supported by National Natural Science Foundation of China under Grant No. 10374075 and Natural Science Foundation of Shanxi Province of China under Grant No. 20001009
文摘The generalized uncertainty relation is introduced to calculate quantum statistic entropy of a black hole. By using the new equation of state density motivated by the generalized uncertainty relation, we discuss entropies of Bose field and Fermi field on the background of the five-dimensional spacetime. In our calculation, we need not introduce cutoff. There is not the divergent logarithmic term as in the original brick-wall method. And it is obtained that the quantum statistic entropy corresponding to black hole horizon is proportional to the area of the horizon. Fhrther it is shown that the entropy of black hole is the entropy of quantum state on the surface of horizon. The black hole＇s entropy is the intrinsic property of the black hole. The entropy is a quantum effect. It makes people further understand the quantum statistic entropy.
基金Project supported by the National Natural Science Foundation of China(Nos.12250410244,11872151)the Jiangsu Province Education Development Special Project-2022 for Double First-ClassSchool Talent Start-up Fund of China(No.2022r109)the Longshan Scholar Program of Jiangsu Province of China。
文摘The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional derivative concept,while the model is solved via the full-spectral method(FSM)and the semi-spectral scheme(SSS).The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques.The SSS is developed by discretizing the time variable,and the space domain is collocated by using equal points.A detailed comparative analysis is made through graphs for various parameters and tables with existing literature.The contour graphs are made to show the behaviors of the velocity and magnetic fields.The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows,and the concept may be extended for variable order models arising in MHD flows.
基金National Natural Science Foundation of China and Cuiying Project of Lanzhou University
文摘We discuss an object from algebraic topology,Hopf invariant,and reinterpret it in terms of the φ-mappingtopological current theory.The main purpose of this paper is to present a new theoretical framework,which can directlygive the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclideanspace R^(2n-1).For the sake of this purpose we introduce a topological tensor current,which can naturally deduce the(n-1)-dimensional topological defect in R^(2n-1) space.If these (n-1)-dimensional topological defects are closed orientedsubmanifolds of R^(2n-1),they are just the (n-1)-dimensional knots.The linking number of these knots is well defined.Using the inner structure of the topological tensor current,the relationship between Hopf invariant and the linkingnumbers of the higher-dimensional knots can be constructed.
文摘ⅠIn the recent years, with the development of the superstring theory (requiring (1+9) dimensional space time) and the application of the Kaluza-Klein theory to the research of the very early phases of the universe (requiring (1+10) dimensional space-time), higher-dimensional physics has assumed a high measure of importance. Emelyanov et al. have in detail evaluated the study situation on the higher-dimensional space-time.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11875200,and 11935009).
文摘We consider both gauged and ungauged minimal supergravities in five dimensions and analyse the charged rotating solutions with two equal angular momenta J.When the electric charge Q∼J^(2/3) with some specific coefficient,we find new extremal black objects emerge that are asymptotic to either Minkowski or global AdS spacetimes and can be best described as degenerate black rings.Their near-horizon geometry is locally AdS3×S^(2),where the periodic U(1)fibre coordinate in S 3 untwists and collapses to be the degenerate part of the AdS3 horizon.It turns out that there are two branches of extremal rotating black holes,starting as the extremal RN black holes of the same mass,but opposite charges.With the increasing of the angular momentum,they will join to become the same degenerate black ring,where the Gibbs free energies however are not continuous at the joining.For the same Q(J)relation,we find that there is in addition a rotating soliton whose mass is smaller than that of the degenerate black ring.
