This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iu +uxx = uv + |u|^2u, vtt-vxx=(|u|^2)xx.First, we prove the existence of a smooth c...This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iu +uxx = uv + |u|^2u, vtt-vxx=(|u|^2)xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].展开更多
In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Lan...In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov展开更多
We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the comp...We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the compactness of minimizing sequence of the constrained minimization problems.展开更多
We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable ...We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.展开更多
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti...In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.展开更多
In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut + auPux + bu2pux + Uzzz = 0 (b 〉 0, p 〉 0). Our results imply that orbital stability of solitary w...In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut + auPux + bu2pux + Uzzz = 0 (b 〉 0, p 〉 0). Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term auPux. For the case of b 〉 0 and 0 〈 p ≤ 2, we obtain that the positive solitary wave Ul(X - ct) is stable when a 〉 0, while that unstable when a 〈 0. The stability for negative solitary wave u2(x - ct) is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p= 2 and a〉0.展开更多
Variational methods are used to study the nonlinear SchrSdinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a co...Variational methods are used to study the nonlinear SchrSdinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.展开更多
In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At firs...In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .展开更多
During dynamic walking of biped robots, the underactuated rotating degree of freedom (DOF) emerges between the support foot and the ground, which makes the biped model hybrid and dimension-variant. This paper addres...During dynamic walking of biped robots, the underactuated rotating degree of freedom (DOF) emerges between the support foot and the ground, which makes the biped model hybrid and dimension-variant. This paper addresses the asymptotic orbit stability for dimension-variant hybrid systems (DVHS). Based on the generalized Poincare map, the stability criterion for DVHS is also presented, and the result is then used to study dynamic walking for a five-link planar biped robot with feet. Time-invariant gait planning and nonlinear control strategy for dynamic walking with fiat feet is also introduced. Simulation results indicate that an asymptotically stable limit cycle of dynamic walking is achieved by the proposed method.展开更多
Coupled trajectory and attitude stability of displaced solar orbits is studied by using sailcraft with a kind of two-folding construction with two unequal rectangular plates forming a right angle. Three-dimensional co...Coupled trajectory and attitude stability of displaced solar orbits is studied by using sailcraft with a kind of two-folding construction with two unequal rectangular plates forming a right angle. Three-dimensional coupled trajectory and attitude equations are developed for the coupled dynamical system, and the results show that all three types of displaced solar orbits widely referenced can be achieved through selecting an appropriate size of the two-folding sail. An anal- ysis of the corresponding linear stability of the trajectory and attitude coupled system is carried out, and both trajectory and attitude linearly stable orbits are found to exist in a small range of parameters, whose non-linear stability is then examined via numerical simulations. Finally, passively stable orbits are found to have weak stability, and such passive means of station-keeping are attractive and useful in practice because of its simplicity.展开更多
Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptib...Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.展开更多
The present paper deals with results of stability/instability of solitary waves with nonzero asymptotic value for a microstructure PDE. By the exact solitary wave solutions and detailed computations, we set up the exp...The present paper deals with results of stability/instability of solitary waves with nonzero asymptotic value for a microstructure PDE. By the exact solitary wave solutions and detailed computations, we set up the explicit expression for the discrimination d′′(c). Finally, a complete study of orbital stablity/instablity for the explicit exact solutions is given.展开更多
In this paper, variable linear feedback control is used to stabilize unstable higher period orbit in nonlinear discrete chaotic dynamical system. The existence of neighborhood in stabilizing higher period orbits is ri...In this paper, variable linear feedback control is used to stabilize unstable higher period orbit in nonlinear discrete chaotic dynamical system. The existence of neighborhood in stabilizing higher period orbits is rigorously proved by functional analysis theory and nonlinear dynamical theory. Numerical simulation is included to support the theoretical analysis in the paper.展开更多
基金supported by the National Natural Science Foundation of China(11401122)Science and technology project of Qufu Normal University(xkj201607)
文摘This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iu +uxx = uv + |u|^2u, vtt-vxx=(|u|^2)xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].
基金The research is supported by the Scientific Research Foundation of Yunnan Provincial Departmentthe Natural Science Foundation of Yunnan Province(No.2005A0026M).
文摘In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov
文摘We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the compactness of minimizing sequence of the constrained minimization problems.
基金National Natural Science Foundation of China(Grants Nos.12271424 and 11871395)National Natural Science Foundation of China(Grants Nos.11971251,11631007 and 12111530003)。
文摘We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.
文摘In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.
基金Supported by the National Natural Science Foundation of China(No.11471215)Innovation Program of Shanghai Municipal Education Commission(No.13ZZ118)+1 种基金Shanghai Leading Academic Discipline Project(No.XTKX2012)Hujiang Foundation of China(No.B14005)
文摘In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut + auPux + bu2pux + Uzzz = 0 (b 〉 0, p 〉 0). Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term auPux. For the case of b 〉 0 and 0 〈 p ≤ 2, we obtain that the positive solitary wave Ul(X - ct) is stable when a 〉 0, while that unstable when a 〈 0. The stability for negative solitary wave u2(x - ct) is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p= 2 and a〉0.
基金supported by the National Natural Science Foundation of China (Nos.10771151 and 10901115)the Scientific Research Fund of Sichuan Provincial Education Department (No.2006A063)the Scientific Research Fund of Science and Technology Bureau of Sichuan Province(No.07JY029-012)
文摘Variational methods are used to study the nonlinear SchrSdinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.
文摘In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .
基金the National Natural Science Foundation of China (No. 50575119)the 863 Program(No. 2006AA04Z253)the Ph.D.Programs Foundation of Ministry of Education of China(No. 20060003026)
文摘During dynamic walking of biped robots, the underactuated rotating degree of freedom (DOF) emerges between the support foot and the ground, which makes the biped model hybrid and dimension-variant. This paper addresses the asymptotic orbit stability for dimension-variant hybrid systems (DVHS). Based on the generalized Poincare map, the stability criterion for DVHS is also presented, and the result is then used to study dynamic walking for a five-link planar biped robot with feet. Time-invariant gait planning and nonlinear control strategy for dynamic walking with fiat feet is also introduced. Simulation results indicate that an asymptotically stable limit cycle of dynamic walking is achieved by the proposed method.
基金supported by the National Natural Science Foundation of China(10832004,10602027)
文摘Coupled trajectory and attitude stability of displaced solar orbits is studied by using sailcraft with a kind of two-folding construction with two unequal rectangular plates forming a right angle. Three-dimensional coupled trajectory and attitude equations are developed for the coupled dynamical system, and the results show that all three types of displaced solar orbits widely referenced can be achieved through selecting an appropriate size of the two-folding sail. An anal- ysis of the corresponding linear stability of the trajectory and attitude coupled system is carried out, and both trajectory and attitude linearly stable orbits are found to exist in a small range of parameters, whose non-linear stability is then examined via numerical simulations. Finally, passively stable orbits are found to have weak stability, and such passive means of station-keeping are attractive and useful in practice because of its simplicity.
文摘Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.
基金Research is supported by Science Foundation of the Education Commission of Beijing(No.KM201210017008)National Natural Science Foundation of China under Grants(No.61403034)Youth Foundation of Beijing Institute of Petrolchemical Technology(No.N10-04)
文摘The present paper deals with results of stability/instability of solitary waves with nonzero asymptotic value for a microstructure PDE. By the exact solitary wave solutions and detailed computations, we set up the explicit expression for the discrimination d′′(c). Finally, a complete study of orbital stablity/instablity for the explicit exact solutions is given.
文摘In this paper, variable linear feedback control is used to stabilize unstable higher period orbit in nonlinear discrete chaotic dynamical system. The existence of neighborhood in stabilizing higher period orbits is rigorously proved by functional analysis theory and nonlinear dynamical theory. Numerical simulation is included to support the theoretical analysis in the paper.