Polarization-dependent second harmonic generation is a widely utilized technique for characterizing symmetry.However,in collinear reflective geometry,the essential beam-splitting device significantly influences both t...Polarization-dependent second harmonic generation is a widely utilized technique for characterizing symmetry.However,in collinear reflective geometry,the essential beam-splitting device significantly influences both the polarization state of the fundamental and harmonic beams,thereby affecting the accuracy of the obtained second-order nonlinear susceptibility.Here,we propose a data correction method to solve this problem to obtain accurate secondorder nonlinear susceptibility.The feasibility and generality of the method are demonstrated through theoretical and experimental validation.展开更多
We theoretically study the effect of Kerr effect on the second-order nonlinearity induced transparency in a double-resonant optical cavity system.We show that in the presence of the Kerr effect,as the strength of the ...We theoretically study the effect of Kerr effect on the second-order nonlinearity induced transparency in a double-resonant optical cavity system.We show that in the presence of the Kerr effect,as the strength of the Kerr effect increases,the absorption curve exhibits an asymmetric-symmetric-asymmetric transition,and the zero absorption point shifts with the increase of the Kerr effect.Furthermore,by changing the strength of the Kerr effect,we can control the width of the transparent window,and the position of the zero-absorption point and meanwhile change the left and right width of the absorption peak.The asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak.The simple dependence of the zeroabsorption point on the strength of Kerr effect suggests that the strength of Kerr effect can be measured by measuring the position of the zero-absorption point in a possible application.展开更多
Continuous control protocols are extensively utilized in traditional MASs,in which information needs to be transmitted among agents consecutively,therefore resulting in excessive consumption of limited resources.To de...Continuous control protocols are extensively utilized in traditional MASs,in which information needs to be transmitted among agents consecutively,therefore resulting in excessive consumption of limited resources.To decrease the control cost,based on ISC,several LFC problems are investigated for second-order MASs without and with time delay,respectively.Firstly,an intermittent sampled controller is designed,and a sufficient and necessary condition is derived,under which state errors between the leader and all the followers approach zero asymptotically.Considering that time delay is inevitable,a new protocol is proposed to deal with the time-delay situation.The error system’s stability is analyzed using the Schur stability theorem,and sufficient and necessary conditions for LFC are obtained,which are closely associated with the coupling gain,the system parameters,and the network structure.Furthermore,for the case where the current position and velocity information are not available,a distributed protocol is designed that depends only on the sampled position information.The sufficient and necessary conditions for LFC are also given.The results show that second-order MASs can achieve the LFC if and only if the system parameters satisfy the inequalities proposed in the paper.Finally,the correctness of the obtained results is verified by numerical simulations.展开更多
In this paper,we investigate the phenomena of electromagnetically induced transparency and the generation of second-order sideband in a Laguerre–Gaussian cavity optorotational system with a Kerr nonlinear medium.Usin...In this paper,we investigate the phenomena of electromagnetically induced transparency and the generation of second-order sideband in a Laguerre–Gaussian cavity optorotational system with a Kerr nonlinear medium.Using the perturbation method,we analyze the first-and second-order sideband generations in the output field from the system under the actions of a strong control field and a weak probe field.Numerical simulations show that the Kerr nonlinearity can lead to the occurrence of the asymmetric line shape in the transmission of the probe field.Comparing with traditional scheme for generating the second-order sideband,our spectral shape of the second-order sideband is amplified and becomes asymmetric,which has potential applications in precision measurement,high-sensitivity devices,and frequency conversion.展开更多
In this paper,the event-triggered consensus control problem for nonlinear uncertain multi-agent systems subject to unknown parameters and external disturbances is considered.The dynamics of subsystems are second-order...In this paper,the event-triggered consensus control problem for nonlinear uncertain multi-agent systems subject to unknown parameters and external disturbances is considered.The dynamics of subsystems are second-order with similar structures,and the nodes are connected by undirected graphs.The event-triggered mechanisms are not only utilized in the transmission of information from the controllers to the actuators,and from the sensors to the controllers within each agent,but also in the communication between agents.Based on the adaptive backstepping method,extra estimators are introduced to handle the unknown parameters,and the measurement errors that occur during the event-triggered communication are well handled by designing compensating terms for the control signals.The presented distributed event-triggered adaptive control laws can guarantee the boundness of the consensus tracking errors and the Zeno behavior is avoided.Meanwhile,the update frequency of the controllers and the load of communication burden are vastly reduced.The obtained control protocol is further applied to a multi-input multi-output second-order nonlinear multi-agent system,and the simulation results show the effectiveness and advantages of our proposed method.展开更多
In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the le...In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.展开更多
Dendrimers are considered as a promising family of organic second-order nonlinear optical （NLO） polymers because of their well-defined structures, easily modified peripheral functional groups, interior branches and ...Dendrimers are considered as a promising family of organic second-order nonlinear optical （NLO） polymers because of their well-defined structures, easily modified peripheral functional groups, interior branches and central cores. In order to obtain NLO materials with high performance, dendrimer structures have been optimized in the past years, such as the ＂branch only＂ and the ＂root containing＂ type dendrimers. This feature article highlights the achievements in exploring the rational design of dendrimers, partially marked by their macroscopic NLO performance.展开更多
A chiral lanthanide metal-organic framework based on enantiopure camphoric acid （D-H2cam）, [Nd3（D-cam）8（H2O）4Cl]n （1）, has been synthesized and characterized by single-crystal X-ray structural analysis, elemen...A chiral lanthanide metal-organic framework based on enantiopure camphoric acid （D-H2cam）, [Nd3（D-cam）8（H2O）4Cl]n （1）, has been synthesized and characterized by single-crystal X-ray structural analysis, elemental analysis, IR, thermal gravimetric, and X-ray powder diffraction. Crystal data for the title compound are as follows： orthorhombic system, space group P212121 with a = 13.8287（7）, b = 14.0715（7）, c = 25.7403（12） A^°, V = 5008.8（4） A^°3, Mr = 1333.08, Z = 4, F（000） = 2644, Dc = 1.768 g/cm^3, μ（MoKα） = 3.189 mm^-1, the final R = 0.0351 and wR = 0.0814 （I 〉 2σ（I））. Compound 1 displays an 8-connected bcu topology 3D framework and hydrogen-bonding interactions stabilize the solid-state structure. The vibrational circular dichroism （VCD） spectrum and second-order nonlinear optical effect of compound 1 have been studied in the solid state.展开更多
The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solut...The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.展开更多
In this study,two new dendronized nonlinear optical(NLO)polymers were synthesized with high FTC chromophore loading density by introduction of high generation chromophore dendrons on the side chains.Due to their suita...In this study,two new dendronized nonlinear optical(NLO)polymers were synthesized with high FTC chromophore loading density by introduction of high generation chromophore dendrons on the side chains.Due to their suitable molecular weights,both of them possessed good solubility in common solvents.They also inherited the advantages of dendrimers(large NLO coefficient),especially for PG2 whose NLO coefficient d33 value was as high as 282 pm·V^–1.Also,PG2 had a good temporal stability with 80%of its maximum value being retained at the temperature as high as 129℃.展开更多
In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contractio...In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.展开更多
The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension ar...The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite展开更多
In this paper, we define an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, desc...In this paper, we define an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of functions. Differentiating these functions twice give second-order nonlinear ODEs that have the defined set of functions as solutions.展开更多
Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generat...Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generated from experiments.Specifically,this framework approximates the latent solution with a deep neural network,which is trained with the constraint of underlying physical laws usually expressed by some equations.In particular,we test the effectiveness of the approach for the Burgers'equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions.The results also indicate that for soliton solutions,the model training costs significantly less time than other initial conditions.展开更多
： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati tech...： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.展开更多
In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are usi...In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.展开更多
This paper studies the leader-following consensus problem for a class of second-order nonlinear multi-agent systems subject to linearly parameterized uncertainty and disturbance. The problem is solved by integrating t...This paper studies the leader-following consensus problem for a class of second-order nonlinear multi-agent systems subject to linearly parameterized uncertainty and disturbance. The problem is solved by integrating the adaptive control technique and the adaptive distributed observer method. The design procedure is illustrated by an example with a group of Van der Pol oscillators as the followers and a harmonic system as the leader.展开更多
In this paper, we study the leader-following rendezvous and flocking problems for a class of second-order nonlinear multi- agent systems, which contain both external disturbances and plant uncertainties. What differs ...In this paper, we study the leader-following rendezvous and flocking problems for a class of second-order nonlinear multi- agent systems, which contain both external disturbances and plant uncertainties. What differs our problems from the conventional leader-following consensus problem is that we need to preserve the connectivity of the communication graph instead of assuming the connectivity of the communication graph. By integrating the adaptive control technique, the distributed observer method and the potential function method, the two problems are both solved. Finally, we apply our results to a group of van der Pol oscillators.展开更多
Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a plane sloping bottom is presented in this paper. For special ease of slope angle β = π/2, th...Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a plane sloping bottom is presented in this paper. For special ease of slope angle β = π/2, this solution can reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline. Interactions between two edge waves including progressive, standing and partially reflected standing waves are also discussed. The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves are also given. The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated, and the corresponding theoreti- cal autocorrelation and spectral density functions of the first and the second orders are derived. The boundary conditions for the determination of the parameters of short edge wave are suggested, that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory. Finally some computation results are demonstrated.展开更多
New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the cond...New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.展开更多
基金This work was supported by the National Natural Science Foundation of China(No.U2230203)the Fundamental Research Funds for the Central Universities.
文摘Polarization-dependent second harmonic generation is a widely utilized technique for characterizing symmetry.However,in collinear reflective geometry,the essential beam-splitting device significantly influences both the polarization state of the fundamental and harmonic beams,thereby affecting the accuracy of the obtained second-order nonlinear susceptibility.Here,we propose a data correction method to solve this problem to obtain accurate secondorder nonlinear susceptibility.The feasibility and generality of the method are demonstrated through theoretical and experimental validation.
基金Supported by the Key Scientific Research Plan of Colleges and Universities in Henan Province(23B140006)the National Natural Science Foundation of China(11965017)。
文摘We theoretically study the effect of Kerr effect on the second-order nonlinearity induced transparency in a double-resonant optical cavity system.We show that in the presence of the Kerr effect,as the strength of the Kerr effect increases,the absorption curve exhibits an asymmetric-symmetric-asymmetric transition,and the zero absorption point shifts with the increase of the Kerr effect.Furthermore,by changing the strength of the Kerr effect,we can control the width of the transparent window,and the position of the zero-absorption point and meanwhile change the left and right width of the absorption peak.The asymmetry absorption curve can be employed to improve the quality factor of the cavity when the frequency detuning is tuned to be around the right peak.The simple dependence of the zeroabsorption point on the strength of Kerr effect suggests that the strength of Kerr effect can be measured by measuring the position of the zero-absorption point in a possible application.
基金supported by the National Natural Science Foundation of China under Grants 62476138 and 42375016.
文摘Continuous control protocols are extensively utilized in traditional MASs,in which information needs to be transmitted among agents consecutively,therefore resulting in excessive consumption of limited resources.To decrease the control cost,based on ISC,several LFC problems are investigated for second-order MASs without and with time delay,respectively.Firstly,an intermittent sampled controller is designed,and a sufficient and necessary condition is derived,under which state errors between the leader and all the followers approach zero asymptotically.Considering that time delay is inevitable,a new protocol is proposed to deal with the time-delay situation.The error system’s stability is analyzed using the Schur stability theorem,and sufficient and necessary conditions for LFC are obtained,which are closely associated with the coupling gain,the system parameters,and the network structure.Furthermore,for the case where the current position and velocity information are not available,a distributed protocol is designed that depends only on the sampled position information.The sufficient and necessary conditions for LFC are also given.The results show that second-order MASs can achieve the LFC if and only if the system parameters satisfy the inequalities proposed in the paper.Finally,the correctness of the obtained results is verified by numerical simulations.
基金supported by the National Natural Science Foundation of China(Grant Nos.12174344 and 12175199)Foundation of Department of Science and Technology of Zhejiang Province(Grant No.2022R52047)。
文摘In this paper,we investigate the phenomena of electromagnetically induced transparency and the generation of second-order sideband in a Laguerre–Gaussian cavity optorotational system with a Kerr nonlinear medium.Using the perturbation method,we analyze the first-and second-order sideband generations in the output field from the system under the actions of a strong control field and a weak probe field.Numerical simulations show that the Kerr nonlinearity can lead to the occurrence of the asymmetric line shape in the transmission of the probe field.Comparing with traditional scheme for generating the second-order sideband,our spectral shape of the second-order sideband is amplified and becomes asymmetric,which has potential applications in precision measurement,high-sensitivity devices,and frequency conversion.
基金supported by National Key R&D Program of China(No.2018YFA0703800)Science Fund for Creative Research Group of the National Natural Science Foundation of China(No.61621002)。
文摘In this paper,the event-triggered consensus control problem for nonlinear uncertain multi-agent systems subject to unknown parameters and external disturbances is considered.The dynamics of subsystems are second-order with similar structures,and the nodes are connected by undirected graphs.The event-triggered mechanisms are not only utilized in the transmission of information from the controllers to the actuators,and from the sensors to the controllers within each agent,but also in the communication between agents.Based on the adaptive backstepping method,extra estimators are introduced to handle the unknown parameters,and the measurement errors that occur during the event-triggered communication are well handled by designing compensating terms for the control signals.The presented distributed event-triggered adaptive control laws can guarantee the boundness of the consensus tracking errors and the Zeno behavior is avoided.Meanwhile,the update frequency of the controllers and the load of communication burden are vastly reduced.The obtained control protocol is further applied to a multi-input multi-output second-order nonlinear multi-agent system,and the simulation results show the effectiveness and advantages of our proposed method.
基金supported by the National Natural Science Foundation of China(61203354)
文摘In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.
基金supported by the National Natural Science Foundation of China(No.21325416)
文摘Dendrimers are considered as a promising family of organic second-order nonlinear optical （NLO） polymers because of their well-defined structures, easily modified peripheral functional groups, interior branches and central cores. In order to obtain NLO materials with high performance, dendrimer structures have been optimized in the past years, such as the ＂branch only＂ and the ＂root containing＂ type dendrimers. This feature article highlights the achievements in exploring the rational design of dendrimers, partially marked by their macroscopic NLO performance.
基金supported by National Natural Science Foundation of China(21401147)Basic Research Program of Natural Science from Shaanxi Provincial Government(2015JQ2032)+2 种基金Scientific Research Program from Education Department of Shaanxi Provincial Government(2013JK0654)Opening Foundation from State Key Laboratory of Coordination Chemistry in Nanjing University(201219)the Program for Distinguished Young Scholars of Xi’an Polytechnic University(201403)
文摘A chiral lanthanide metal-organic framework based on enantiopure camphoric acid （D-H2cam）, [Nd3（D-cam）8（H2O）4Cl]n （1）, has been synthesized and characterized by single-crystal X-ray structural analysis, elemental analysis, IR, thermal gravimetric, and X-ray powder diffraction. Crystal data for the title compound are as follows： orthorhombic system, space group P212121 with a = 13.8287（7）, b = 14.0715（7）, c = 25.7403（12） A^°, V = 5008.8（4） A^°3, Mr = 1333.08, Z = 4, F（000） = 2644, Dc = 1.768 g/cm^3, μ（MoKα） = 3.189 mm^-1, the final R = 0.0351 and wR = 0.0814 （I 〉 2σ（I））. Compound 1 displays an 8-connected bcu topology 3D framework and hydrogen-bonding interactions stabilize the solid-state structure. The vibrational circular dichroism （VCD） spectrum and second-order nonlinear optical effect of compound 1 have been studied in the solid state.
文摘The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.
基金financially supported by the National Natural Science Foundation of China (No. 21734007)
文摘In this study,two new dendronized nonlinear optical(NLO)polymers were synthesized with high FTC chromophore loading density by introduction of high generation chromophore dendrons on the side chains.Due to their suitable molecular weights,both of them possessed good solubility in common solvents.They also inherited the advantages of dendrimers(large NLO coefficient),especially for PG2 whose NLO coefficient d33 value was as high as 282 pm·V^–1.Also,PG2 had a good temporal stability with 80%of its maximum value being retained at the temperature as high as 129℃.
基金Supported by the Scientific Research Fund of Education Department of Hunan Province(07C680)
文摘In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.
基金Project supported by the National Natural Science Foundation of China(Grant No.10926082)the Natural Science Foundation of Anhui Province of China(Grant No.KJ2010A128)the Fund for Youth of Anhui Normal University,China(Grant No.2009xqn55)
文摘The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite
文摘In this paper, we define an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of functions. Differentiating these functions twice give second-order nonlinear ODEs that have the defined set of functions as solutions.
基金supported by the National Natural Science Foundation of China(No.11675054)Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)Science and Technology Commission of Shanghai Municipality(No.18dz2271000)。
文摘Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generated from experiments.Specifically,this framework approximates the latent solution with a deep neural network,which is trained with the constraint of underlying physical laws usually expressed by some equations.In particular,we test the effectiveness of the approach for the Burgers'equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions.The results also indicate that for soliton solutions,the model training costs significantly less time than other initial conditions.
基金Supported by the Scientific Research Fund of Hunan Provincial Education Department(09A082)
文摘： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.
文摘In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.
文摘This paper studies the leader-following consensus problem for a class of second-order nonlinear multi-agent systems subject to linearly parameterized uncertainty and disturbance. The problem is solved by integrating the adaptive control technique and the adaptive distributed observer method. The design procedure is illustrated by an example with a group of Van der Pol oscillators as the followers and a harmonic system as the leader.
文摘In this paper, we study the leader-following rendezvous and flocking problems for a class of second-order nonlinear multi- agent systems, which contain both external disturbances and plant uncertainties. What differs our problems from the conventional leader-following consensus problem is that we need to preserve the connectivity of the communication graph instead of assuming the connectivity of the communication graph. By integrating the adaptive control technique, the distributed observer method and the potential function method, the two problems are both solved. Finally, we apply our results to a group of van der Pol oscillators.
文摘Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a plane sloping bottom is presented in this paper. For special ease of slope angle β = π/2, this solution can reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline. Interactions between two edge waves including progressive, standing and partially reflected standing waves are also discussed. The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves are also given. The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated, and the corresponding theoreti- cal autocorrelation and spectral density functions of the first and the second orders are derived. The boundary conditions for the determination of the parameters of short edge wave are suggested, that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory. Finally some computation results are demonstrated.
文摘New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.
