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
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
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
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, 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
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
A series of CdS nanoparticles with different surfaces were prepared by colloidal chemical method and reverse micelle method. Their second-order nonlinear optical (NLO) properties were experimentally studied in solutio...A series of CdS nanoparticles with different surfaces were prepared by colloidal chemical method and reverse micelle method. Their second-order nonlinear optical (NLO) properties were experimentally studied in solution by newly developed hyper-Rayleigh scattering (HRS) technique. The results show that 'per particle' first-order hyperpolarizability beta values are sensitive To the synthetic method and the surface chemical modification.展开更多
This paper demonstrates the second-order nonlinear hyperpolarisability γ of all-trans-β-carotene in different solvents by linear spectroscopic technique that is based on resonance Raman scattering and UV-VIS （Ultra...This paper demonstrates the second-order nonlinear hyperpolarisability γ of all-trans-β-carotene in different solvents by linear spectroscopic technique that is based on resonance Raman scattering and UV-VIS （Ultraviolet-visible） absorption spectroscopy. Owing to the two-level model well describing the link that exists between the resonance Raman scattering and stimulated Raman scattering, the stimulated Raman polarisability αR can be calculated through the two-photon resonance system. The value of γ of all-trans-β-carotene in carbon bisulfide solution is 6.435×10^-33 esu （1 esu of resistance = 8.98755×10^11Ω） that is close to the true value, because the solution of all-trans-β-carotene in carbon bisulfide satisfies the rigid resonance Raman scattering condition. This method is expected to be worthy of applications to measure the second-order nonlinear hyperpolaxisability of a conjugate organic molecule.展开更多
In this paper,we investigate the photon correlations and the statistical properties of light produced by an optical cavity with an embedded quantum well interacting with squeezed light.We show that the squeezed source...In this paper,we investigate the photon correlations and the statistical properties of light produced by an optical cavity with an embedded quantum well interacting with squeezed light.We show that the squeezed source substantially improves the intensity of the emitted light and generates a narrowing and a duplication of the spectrum peaks.With a strong dependence on frequency detuning,the cavity produces considerably squeezed radiation,and perfect squeezing is predicted for weak light–matter interactions.Furthermore,the system under consideration presents a bunching effect of the transmitted radiation resulting from weak pumping of the coherent field.The results obtained may have potential applications in the fields of very accurate measurement and quantum computing.展开更多
The design and the synthesis of two conjugated donor acceptor imidazole derivatives(1, 2) were carried out for second order nonlinear optics. The thermal properties, the transparency and second order nonlinear opti...The design and the synthesis of two conjugated donor acceptor imidazole derivatives(1, 2) were carried out for second order nonlinear optics. The thermal properties, the transparency and second order nonlinear optical properties of the molecules were investigated. The experimental results indicate that a good nonlinearity transparency thermal stability trade off is achieved for them.展开更多
The unconventional photon blockade(UPB)for low-frequency mode is investigated in a three-mode system with double second-order nonlinearity.By analyzing the Hamiltonian of the system,the optimal analytic condition of U...The unconventional photon blockade(UPB)for low-frequency mode is investigated in a three-mode system with double second-order nonlinearity.By analyzing the Hamiltonian of the system,the optimal analytic condition of UPB in low-frequency mode is obtained.The numerical results are calculated by solving the master equation in a truncated Fock space,which agrees well with the analytic conditions.Through the numerical analysis of the system,it is found that the weak driving strength is favorable for the system to realize the UPB effect,and the system is insensitive to the changes of attenuation rate and environmental temperature.The comparison with the two-mode system and another similar threemode system shows that,under similar system parameters,the UPB effect of this double two-order nonlinear system is more obvious.展开更多
In this paper, we define 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 will also define a group of solutions x(t) that are ...In this paper, we define 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 will also define a group of solutions x(t) that are equal to the amplitude. This is a generalized amplitude function. We are using Abel’s methods, described by Armitage and Eberlein. And finally, we define an exponential function whose exponent is the product of a complex number and the upper limit of integration in a non-elementary integral that can be arbitrary. At least three groups of non-elementary functions are special cases of this complex function.展开更多
In this paper, we define four new examples of the non-elementary expo-elliptic functions. This is an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elem...In this paper, we define four new examples of the non-elementary expo-elliptic functions. This is 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. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles, and systems of nonlinear ODEs that these functions are giving solutions to.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
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
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
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 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(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.
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℃.
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.
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.
基金National Natural Science Foundation of China! (No.59582005)
文摘A series of CdS nanoparticles with different surfaces were prepared by colloidal chemical method and reverse micelle method. Their second-order nonlinear optical (NLO) properties were experimentally studied in solution by newly developed hyper-Rayleigh scattering (HRS) technique. The results show that 'per particle' first-order hyperpolarizability beta values are sensitive To the synthetic method and the surface chemical modification.
基金Project supported by the National Natural Science Foundation of China (Gant Nos. 10774057 and 10974067)
文摘This paper demonstrates the second-order nonlinear hyperpolarisability γ of all-trans-β-carotene in different solvents by linear spectroscopic technique that is based on resonance Raman scattering and UV-VIS （Ultraviolet-visible） absorption spectroscopy. Owing to the two-level model well describing the link that exists between the resonance Raman scattering and stimulated Raman scattering, the stimulated Raman polarisability αR can be calculated through the two-photon resonance system. The value of γ of all-trans-β-carotene in carbon bisulfide solution is 6.435×10^-33 esu （1 esu of resistance = 8.98755×10^11Ω） that is close to the true value, because the solution of all-trans-β-carotene in carbon bisulfide satisfies the rigid resonance Raman scattering condition. This method is expected to be worthy of applications to measure the second-order nonlinear hyperpolaxisability of a conjugate organic molecule.
文摘In this paper,we investigate the photon correlations and the statistical properties of light produced by an optical cavity with an embedded quantum well interacting with squeezed light.We show that the squeezed source substantially improves the intensity of the emitted light and generates a narrowing and a duplication of the spectrum peaks.With a strong dependence on frequency detuning,the cavity produces considerably squeezed radiation,and perfect squeezing is predicted for weak light–matter interactions.Furthermore,the system under consideration presents a bunching effect of the transmitted radiation resulting from weak pumping of the coherent field.The results obtained may have potential applications in the fields of very accurate measurement and quantum computing.
基金Supported by the Natural Science Foundation of Hubei ProvinceChina(No.2 0 0 0 J15 6 )
文摘The design and the synthesis of two conjugated donor acceptor imidazole derivatives(1, 2) were carried out for second order nonlinear optics. The thermal properties, the transparency and second order nonlinear optical properties of the molecules were investigated. The experimental results indicate that a good nonlinearity transparency thermal stability trade off is achieved for them.
基金Project supported by the National Natural Science Foundation of China(Grant No.11647054)the Natural Science Foundation of Jilin Province,China(Grant No.JJKH20181088KJ)。
文摘The unconventional photon blockade(UPB)for low-frequency mode is investigated in a three-mode system with double second-order nonlinearity.By analyzing the Hamiltonian of the system,the optimal analytic condition of UPB in low-frequency mode is obtained.The numerical results are calculated by solving the master equation in a truncated Fock space,which agrees well with the analytic conditions.Through the numerical analysis of the system,it is found that the weak driving strength is favorable for the system to realize the UPB effect,and the system is insensitive to the changes of attenuation rate and environmental temperature.The comparison with the two-mode system and another similar threemode system shows that,under similar system parameters,the UPB effect of this double two-order nonlinear system is more obvious.
文摘In this paper, we define 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 will also define a group of solutions x(t) that are equal to the amplitude. This is a generalized amplitude function. We are using Abel’s methods, described by Armitage and Eberlein. And finally, we define an exponential function whose exponent is the product of a complex number and the upper limit of integration in a non-elementary integral that can be arbitrary. At least three groups of non-elementary functions are special cases of this complex function.
文摘In this paper, we define four new examples of the non-elementary expo-elliptic functions. This is 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. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles, and systems of nonlinear ODEs that these functions are giving solutions to.
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
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.
