It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instan...It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.展开更多
In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.
We show that a class of spectral problems are related to the spectral problem of the Volterra lattice through a gauge transformation. The transformation is given. We hope that our discussion can draw attention to the ...We show that a class of spectral problems are related to the spectral problem of the Volterra lattice through a gauge transformation. The transformation is given. We hope that our discussion can draw attention to the study of gauge transformation theory of differential-difference integrable systems.展开更多
When the motion of a particle is constrained, excess terms exist using hermitian form of Cartesian momentum pi （i=1, 2,3） in usual kinetic energy （1/2/μ）∑ pi^2 , and the correct kinetic energy turns out to be （...When the motion of a particle is constrained, excess terms exist using hermitian form of Cartesian momentum pi （i=1, 2,3） in usual kinetic energy （1/2/μ）∑ pi^2 , and the correct kinetic energy turns out to be （1/2μ） ∑1/ fipi f ipi, where fi are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper the explicit form of the dummy functions fi is given for a charged rigid planar rotator in the uniform magnetic field with different gauge chosen. Under different gauges, we have different sets of dummy factors. It means that these factors do not have direct observable effect.展开更多
The author of this paper has put forward a unified program of gauge field from the mathematical and physical picture of the principal associated bundles: thinking that our universe may have more fundamental interactio...The author of this paper has put forward a unified program of gauge field from the mathematical and physical picture of the principal associated bundles: thinking that our universe may have more fundamental interactions than the four fundamental interactions, and these basic interaction gauge fields are only the projection components to the base manifold, that is our universe, from a unified gauge potential or connection of the principal associated bundle manifold on the base manifold. These components can satisfy the transformation of gauge potential, and can even be transformed from one basic interaction gauge potential to another basic interaction gauge potential, and can be summarized into a unified equation, that is, the generalized gauge Equation (GGE), but the gauge potential or connection on the principal bundle is invariant, corresponding to the invariance of gauge transformation [1]. In this paper, we will continue to discuss this aspect concretely, and specifically construct a spatiotemporal model with the frame bundle as the principal bundle, and the tensor bundle as the associated bundle, so that the four fundamental interactions, especially the electromagnetic interaction and the gravitational interaction, can be reflected in the bottom manifold, that is, the regional distributions in our universe. Furthermore, this paper studies the existence of gauge transformation across basic interactions by establishing a model of gauge transformation of basic interaction field;it is found that the unified expression formula is GGE and the expression relation on the curvature of space-time. Therefore, the author discusses the feasibility of the generalized gauge transformation across the basic electromagnetic interaction and the basic gravitational interaction, and on this basis, specifically determines a method or way to find the generalized gauge transformation, so as to try to realize the last step of the “unification” of the four fundamental interactions in physics, that is, the “unification” of electromagnetism and gravity.展开更多
This paper attempts to propose a grand unified guiding principle of gauge fields from the mathematical and physical picture of fiber bundles: it is believed that our universe may have more fundamental interactions tha...This paper attempts to propose a grand unified guiding principle of gauge fields from the mathematical and physical picture of fiber bundles: it is believed that our universe may have more fundamental interactions than the four fundamental interactions, and the gauge fields of these fundamental interactions are just a unified gauge potential on the fiber bundle manifold or the components connected to the bottom manifold, that is, our universe;these components can meet the transformation of gauge potential, and even can be transformed from a fundamental interaction gauge potential to another fundamental interaction gauge potential, and can be summarized into a unified equation, namely the expression of the generalized gauge equation, corresponding to the gauge transformation invariance;so gauge transformation invariance is a necessary condition to unify field theory, but quantization of field is not a necessary condition;the four (or more) fundamental interaction fields of the universe are unified into a universal gauge field defined by the connection of the principal fiber bundle on the cosmic base manifold.展开更多
Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transform...Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.展开更多
This paper proposes a model-based control framework for vehicle platooning systems with secondorder nonlinear dynamics operating over switching signed networks,time-varying delays,and deception attacks.The study inclu...This paper proposes a model-based control framework for vehicle platooning systems with secondorder nonlinear dynamics operating over switching signed networks,time-varying delays,and deception attacks.The study includes two configurations:a leaderless structure using Finite-Time Non-Singular Terminal Bipartite Consensus(FNTBC)and Fixed-Time Bipartite Consensus(FXTBC),and a leader—follower structure ensuring structural balance and robustness against deceptive signals.In the leaderless model,a bipartite controller based on impulsive control theory,gauge transformation,and Markovian switching Lyapunov functions ensures mean-square stability and coordination under deception attacks and communication delays.The FNTBC achieves finite-time convergence depending on initial conditions,while the FXTBC guarantees fixed-time convergence independent of them,providing adaptability to different operating states.In the leader—follower case,a discontinuous impulsive control law synchronizes all followers with the leader despite deceptive attacks and switching topologies,maintaining robust coordination through nonlinear corrective mechanisms.To validate the approach,simulations are conducted on systems of five and seventeen vehicles in both leaderless and leader—follower configurations.The results demonstrate that the proposed framework achieves rapid consensus,strong robustness,and high resistance to deception attacks,offering a secure and scalable model-based control solution for modern vehicular communication networks.展开更多
This paper is concerned with the construction of two types of generalized Heisenberg supermagnet model with the constraint S^(3)=S,including the inhomogeneous Heisenberg supermagnet model in(1+1)dimensions and the(2+1...This paper is concerned with the construction of two types of generalized Heisenberg supermagnet model with the constraint S^(3)=S,including the inhomogeneous Heisenberg supermagnet model in(1+1)dimensions and the(2+1)-dimensional Heisenberg supermagnet model.Furthermore,by means of the gauge transformation,we investigate the gauge equivalent counterparts,which are the(1+1)-dimensional inhomogeneous nonlinear Schrodinger equation and the(2+1)-dimensional super nonlinear Schrodinger equation,respectively.展开更多
Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the stand...Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.展开更多
We derive the adiabatic and non-adiabatic Berry phases in the generalized Jaynes-Cummings model of multi-photon process. The results show that the adiabatic Berry phase is kept a constant π independent of all the par...We derive the adiabatic and non-adiabatic Berry phases in the generalized Jaynes-Cummings model of multi-photon process. The results show that the adiabatic Berry phase is kept a constant π independent of all the parameters, while the non-adiabatic approximate Berry phase is parameter-dependent, proportional to the average photon number m, and tends to be constant with the increasing detuning. In the ease of exact n-photon resonance and an integer ratio of m/n, the two results coincide with each other, otherwise there appears an additional non-trivial phase factor.展开更多
In this paper, we investigate the Berry phase of two coupled arbitrary spins driven by a time-varying magnetic field where the Hamiltonian is explicitly tlme-dependent. Using a technique of time-dependent gauge transf...In this paper, we investigate the Berry phase of two coupled arbitrary spins driven by a time-varying magnetic field where the Hamiltonian is explicitly tlme-dependent. Using a technique of time-dependent gauge transform the Berry phase and time-evolution operator are found explicitly in the adiabatic approximation. The general solutions for arbitrary spins are applied to the spin-1/2 system as an example of explanation.展开更多
Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isos...Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.展开更多
Conservation law plays a very important role in many geometric variational problems and related elliptic systems.In this note,we refine the conservation law obtained by Lamm-Rivière for fourth order systems and d...Conservation law plays a very important role in many geometric variational problems and related elliptic systems.In this note,we refine the conservation law obtained by Lamm-Rivière for fourth order systems and de Longueville-Gastel for general even order systems.展开更多
An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density|u|^(2)is conserved.We also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where...An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density|u|^(2)is conserved.We also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where the total particle density∑^(n)_(j)=_(1)∣u_(j)∣^(2)is conserved.These equations are related to nonisospectral scalar and vector nonlinear Schrödinger equations.Infinitely many conservation laws are obtained.Gauge transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations,both scalar and vector cases are derived.Solutions and dynamics are analyzed and illustrated.Some solutions exhibit features of localized-like waves.展开更多
Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge...Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.展开更多
Based on the exact solution of the time-dependent Schrodinger equation for two-species Bose-Einstein condensates （BECs） consisting of two hyperfine states of the atoms coupled by a tuned adiabatic and time-varying R...Based on the exact solution of the time-dependent Schrodinger equation for two-species Bose-Einstein condensates （BECs） consisting of two hyperfine states of the atoms coupled by a tuned adiabatic and time-varying Raman coupling, we obtain analytically the entanglement dynamics of the system with various initial states, particularly the SU（2） coherent state, for both of cases with and without the nonlinear interactions. It is shown that the effect of nonlinear interaction on the entanglement appears only in a longer time period depending on the BEC parameters.展开更多
There are two kinds of definitions of perturbation of physical quantities in the framework of general relativity： one is direct, the other is geometrical. Correspondingly, there are two types of gauge transformation ...There are two kinds of definitions of perturbation of physical quantities in the framework of general relativity： one is direct, the other is geometrical. Correspondingly, there are two types of gauge transformation related with these two definitions. The passive approach is based on the property of general covariance, and the active one is through the action of Lie-derivative. Although under a proper coordinate choice, the two approaches seem to agree with each other, they are different in nature. The geometrical definition of relativistic perturbation and the active approach for gauge transformation are more rigorous in mathematics and less confusing in physical explanation. The direct definition, however, seems to be plagued with difficulties in physical meaning, and the passive approach is more awkward to use, especially for high-order gauge transformations.展开更多
The gauge invariance of the electromagnetic field in gravitational field is an important question. We prove d' Alembert equation in gravitational field with gauge invariance under the Lorentz condition. Using the ...The gauge invariance of the electromagnetic field in gravitational field is an important question. We prove d' Alembert equation in gravitational field with gauge invariance under the Lorentz condition. Using the kinematic equation of photon in normal static and spherically symmetric gravitational fields, we deduce the orbital equation of photon. As a special example, we explicate the deduction and discussion about the deviation angular of light in Reissner-Nordstrom space-time.展开更多
We introduce a new integrable model to investigate the dynamics of two component quasi-particle condensates with spatiotemporal interaction strengths. We derive the associated Lax pair of the coupled Gross-Pitaevskii ...We introduce a new integrable model to investigate the dynamics of two component quasi-particle condensates with spatiotemporal interaction strengths. We derive the associated Lax pair of the coupled Gross-Pitaevskii （GP） equation and construct matter wave solitons, We show that the spatiotemporal binary interaction strengths not only facilitate the stabilization of the condensates, but also enables one to fabricate condensates with desirable densities, geometries, and properties, leading to the so-called ＂designer quasi-particle condensates＂.展开更多
文摘It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.
基金supported by the Fundamental Research Funds for the Central Universities(2015QNA43)
文摘In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.
基金Supported by the National Natural Science Foundation of China under Grant No 11371241
文摘We show that a class of spectral problems are related to the spectral problem of the Volterra lattice through a gauge transformation. The transformation is given. We hope that our discussion can draw attention to the study of gauge transformation theory of differential-difference integrable systems.
基金The project supported by the Program for New Century Excellent Talents in Universities, Ministry of Education, and the Key Teaching Reform Program of Hunan Province
文摘When the motion of a particle is constrained, excess terms exist using hermitian form of Cartesian momentum pi （i=1, 2,3） in usual kinetic energy （1/2/μ）∑ pi^2 , and the correct kinetic energy turns out to be （1/2μ） ∑1/ fipi f ipi, where fi are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper the explicit form of the dummy functions fi is given for a charged rigid planar rotator in the uniform magnetic field with different gauge chosen. Under different gauges, we have different sets of dummy factors. It means that these factors do not have direct observable effect.
文摘The author of this paper has put forward a unified program of gauge field from the mathematical and physical picture of the principal associated bundles: thinking that our universe may have more fundamental interactions than the four fundamental interactions, and these basic interaction gauge fields are only the projection components to the base manifold, that is our universe, from a unified gauge potential or connection of the principal associated bundle manifold on the base manifold. These components can satisfy the transformation of gauge potential, and can even be transformed from one basic interaction gauge potential to another basic interaction gauge potential, and can be summarized into a unified equation, that is, the generalized gauge Equation (GGE), but the gauge potential or connection on the principal bundle is invariant, corresponding to the invariance of gauge transformation [1]. In this paper, we will continue to discuss this aspect concretely, and specifically construct a spatiotemporal model with the frame bundle as the principal bundle, and the tensor bundle as the associated bundle, so that the four fundamental interactions, especially the electromagnetic interaction and the gravitational interaction, can be reflected in the bottom manifold, that is, the regional distributions in our universe. Furthermore, this paper studies the existence of gauge transformation across basic interactions by establishing a model of gauge transformation of basic interaction field;it is found that the unified expression formula is GGE and the expression relation on the curvature of space-time. Therefore, the author discusses the feasibility of the generalized gauge transformation across the basic electromagnetic interaction and the basic gravitational interaction, and on this basis, specifically determines a method or way to find the generalized gauge transformation, so as to try to realize the last step of the “unification” of the four fundamental interactions in physics, that is, the “unification” of electromagnetism and gravity.
文摘This paper attempts to propose a grand unified guiding principle of gauge fields from the mathematical and physical picture of fiber bundles: it is believed that our universe may have more fundamental interactions than the four fundamental interactions, and the gauge fields of these fundamental interactions are just a unified gauge potential on the fiber bundle manifold or the components connected to the bottom manifold, that is, our universe;these components can meet the transformation of gauge potential, and even can be transformed from a fundamental interaction gauge potential to another fundamental interaction gauge potential, and can be summarized into a unified equation, namely the expression of the generalized gauge equation, corresponding to the gauge transformation invariance;so gauge transformation invariance is a necessary condition to unify field theory, but quantization of field is not a necessary condition;the four (or more) fundamental interaction fields of the universe are unified into a universal gauge field defined by the connection of the principal fiber bundle on the cosmic base manifold.
文摘Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.
基金Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP.2/103/46”Deanship of Scientific Research at Northern Border University,Arar,Saudi Arabia for funding this research work through project number“NBU-FFR-2025-871-15”funding from Prince Sattam bin Abdulaziz University project number(PSAU/2025/R/1447).
文摘This paper proposes a model-based control framework for vehicle platooning systems with secondorder nonlinear dynamics operating over switching signed networks,time-varying delays,and deception attacks.The study includes two configurations:a leaderless structure using Finite-Time Non-Singular Terminal Bipartite Consensus(FNTBC)and Fixed-Time Bipartite Consensus(FXTBC),and a leader—follower structure ensuring structural balance and robustness against deceptive signals.In the leaderless model,a bipartite controller based on impulsive control theory,gauge transformation,and Markovian switching Lyapunov functions ensures mean-square stability and coordination under deception attacks and communication delays.The FNTBC achieves finite-time convergence depending on initial conditions,while the FXTBC guarantees fixed-time convergence independent of them,providing adaptability to different operating states.In the leader—follower case,a discontinuous impulsive control law synchronizes all followers with the leader despite deceptive attacks and switching topologies,maintaining robust coordination through nonlinear corrective mechanisms.To validate the approach,simulations are conducted on systems of five and seventeen vehicles in both leaderless and leader—follower configurations.The results demonstrate that the proposed framework achieves rapid consensus,strong robustness,and high resistance to deception attacks,offering a secure and scalable model-based control solution for modern vehicular communication networks.
基金supported by the National Natural Science Foundation of China(Grant Nos.12461048 and 12061051)Natural Science Foundation of Inner Mongolia Autonomous Region(Grant No.2023MS01003)+2 种基金the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(Grant No.NJYT23096)the financial support from the Program of China Scholarships Council(Grant No.202306810054)for one year's study at the University of Leedsthe support of Professor Ke Wu and Professor Weizhong Zhao at Capital Normal University,China。
文摘This paper is concerned with the construction of two types of generalized Heisenberg supermagnet model with the constraint S^(3)=S,including the inhomogeneous Heisenberg supermagnet model in(1+1)dimensions and the(2+1)-dimensional Heisenberg supermagnet model.Furthermore,by means of the gauge transformation,we investigate the gauge equivalent counterparts,which are the(1+1)-dimensional inhomogeneous nonlinear Schrodinger equation and the(2+1)-dimensional super nonlinear Schrodinger equation,respectively.
基金the National Natural Science Foundation of China(Grant Nos.11601312,11631007,and 11875040)
文摘Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.
基金Supported by the National Natural Science Foundation of China under Grants Nos.11075099,11047167,and 11105087
文摘We derive the adiabatic and non-adiabatic Berry phases in the generalized Jaynes-Cummings model of multi-photon process. The results show that the adiabatic Berry phase is kept a constant π independent of all the parameters, while the non-adiabatic approximate Berry phase is parameter-dependent, proportional to the average photon number m, and tends to be constant with the increasing detuning. In the ease of exact n-photon resonance and an integer ratio of m/n, the two results coincide with each other, otherwise there appears an additional non-trivial phase factor.
基金Project supported by the National Natural Science Foundation of China （Grant No 10475053）
文摘In this paper, we investigate the Berry phase of two coupled arbitrary spins driven by a time-varying magnetic field where the Hamiltonian is explicitly tlme-dependent. Using a technique of time-dependent gauge transform the Berry phase and time-evolution operator are found explicitly in the adiabatic approximation. The general solutions for arbitrary spins are applied to the spin-1/2 system as an example of explanation.
基金supported by the National Natural Science Foundation of China under Grant Nos. 10371070 and 10671121the Foundation for Excellent Postgraduates of Shanghai University under Grant No. Shucx080127
文摘Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.
基金supported by the Young Scientist Program of the Ministry of Science and Technology of China(2021YFA1002200)the National Natural Science Foundation of China(12101362)+4 种基金the Natural Science Foundation of Shandong Province(ZR2021QA003)supported by the National Natural Science Foundation of China(12271296)the Natural Science Foundation of Hubei Province(2024AFA061)supported by the National Natural Science Foundation of China(11571131)the Open Research Fund of Key Laboratory of Nonlinear Analysis&Applications(Central China Normal University),Ministry of Education,P.R.China。
文摘Conservation law plays a very important role in many geometric variational problems and related elliptic systems.In this note,we refine the conservation law obtained by Lamm-Rivière for fourth order systems and de Longueville-Gastel for general even order systems.
基金supported by the NSF of China (Nos. 11 875 040, 12 126 352, 12 126 343)。
文摘An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density|u|^(2)is conserved.We also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where the total particle density∑^(n)_(j)=_(1)∣u_(j)∣^(2)is conserved.These equations are related to nonisospectral scalar and vector nonlinear Schrödinger equations.Infinitely many conservation laws are obtained.Gauge transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations,both scalar and vector cases are derived.Solutions and dynamics are analyzed and illustrated.Some solutions exhibit features of localized-like waves.
基金the Science Fundation for Young Teachers of Southwest University(No.SWUQ2006028)
文摘Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.
基金Project supported by the National Natural Science Foundation of China （Grant No 10475053）.
文摘Based on the exact solution of the time-dependent Schrodinger equation for two-species Bose-Einstein condensates （BECs） consisting of two hyperfine states of the atoms coupled by a tuned adiabatic and time-varying Raman coupling, we obtain analytically the entanglement dynamics of the system with various initial states, particularly the SU（2） coherent state, for both of cases with and without the nonlinear interactions. It is shown that the effect of nonlinear interaction on the entanglement appears only in a longer time period depending on the BEC parameters.
基金supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,the Ministry of Educationthe Chinese Academy of Sciences under Grant No.KJCX3-SYW-N2
文摘There are two kinds of definitions of perturbation of physical quantities in the framework of general relativity： one is direct, the other is geometrical. Correspondingly, there are two types of gauge transformation related with these two definitions. The passive approach is based on the property of general covariance, and the active one is through the action of Lie-derivative. Although under a proper coordinate choice, the two approaches seem to agree with each other, they are different in nature. The geometrical definition of relativistic perturbation and the active approach for gauge transformation are more rigorous in mathematics and less confusing in physical explanation. The direct definition, however, seems to be plagued with difficulties in physical meaning, and the passive approach is more awkward to use, especially for high-order gauge transformations.
文摘The gauge invariance of the electromagnetic field in gravitational field is an important question. We prove d' Alembert equation in gravitational field with gauge invariance under the Lorentz condition. Using the kinematic equation of photon in normal static and spherically symmetric gravitational fields, we deduce the orbital equation of photon. As a special example, we explicate the deduction and discussion about the deviation angular of light in Reissner-Nordstrom space-time.
基金Project supported by the UGC,DAE-NBHM,and DST,Government of India
文摘We introduce a new integrable model to investigate the dynamics of two component quasi-particle condensates with spatiotemporal interaction strengths. We derive the associated Lax pair of the coupled Gross-Pitaevskii （GP） equation and construct matter wave solitons, We show that the spatiotemporal binary interaction strengths not only facilitate the stabilization of the condensates, but also enables one to fabricate condensates with desirable densities, geometries, and properties, leading to the so-called ＂designer quasi-particle condensates＂.
