By utilizing a fixed point theorem on cone, some new results on the existence ofpositive periodic solutions for nonautonomous differential equations with delay are derived.
This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact ...This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.展开更多
The long-time behaviour of a two-dimensional nonautonomous nonlinear Schrodinger equation is considered. The existence! of uniform attractor is proved and the upper bound of the uniform attractor's Housdorff dimen...The long-time behaviour of a two-dimensional nonautonomous nonlinear Schrodinger equation is considered. The existence! of uniform attractor is proved and the upper bound of the uniform attractor's Housdorff dimension is given.展开更多
In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is studied. A family of convergent approximate inertial manifolds for a class of evolution equations has been constructed w...In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is studied. A family of convergent approximate inertial manifolds for a class of evolution equations has been constructed when the spectral gap condition is satisfied.展开更多
New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous differ...New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous difference equations is negative definite. In this paper, we consider the case that the Liapunov function is only positive definite and its variation is semi-negative definite. At these weaker conditions, we put forward a new asymptotical stability theorem of nonautonomous difference equations by adding to extra conditions on the variation. After that, in addition to the hypotheses of our new asymptotical stability theorem, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations provided that the Liapunov function has an indefinitely small upper bound. Example is given to verify our results in the last.展开更多
The existence and the behavior of solutions to the differential-iterative equation arestudied without the restriction that f is monotone An error in and available paper is correctedhere.
A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarit...A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.展开更多
The global attractivity of the delay difference equation Deltax(n) + a(n)x(n) + f(n, Sigma(s = -k)(0)(q) over bar (s), (n) x(s + n)) = 0, which includes the discrete type of many mathematical ecological equations, was...The global attractivity of the delay difference equation Deltax(n) + a(n)x(n) + f(n, Sigma(s = -k)(0)(q) over bar (s), (n) x(s + n)) = 0, which includes the discrete type of many mathematical ecological equations, was discussed. The sufficient conditions that guarantee every solution to converge to zero are obtained. Many known results are improved and generated.展开更多
In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomou...In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomous evolution equations.展开更多
We present exact bright multi-soliton solutions of a generalized nonautonomous nonlinear Schroinger equation with time-and space-dependent distributed coefficients and an external potential which describes a pulse pro...We present exact bright multi-soliton solutions of a generalized nonautonomous nonlinear Schroinger equation with time-and space-dependent distributed coefficients and an external potential which describes a pulse propagating in nonlinear media when its transverse and longitudinal directions are nonuniformly distributed.Such solutions exist in certain constraint conditions on the coefficients depicting dispersion,nonlinearity,and gain（loss）.Various shapes of bright solitons and interesting interactions between two solitons are observed.Physical applications of interest to the field and stability of the solitons are discussed.展开更多
Under the assumption that h(z) is strictly monotone the existence of solutions to a type of nonlinear differential-iterative equations in the form of x’(t) = g(x(t)) -h(x(x(t))) is discussed according to the behavior...Under the assumption that h(z) is strictly monotone the existence of solutions to a type of nonlinear differential-iterative equations in the form of x’(t) = g(x(t)) -h(x(x(t))) is discussed according to the behavior of the quasi-isoclinic curve C: x=h^-1 (g(t))展开更多
This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay dif...This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential equations.We prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square sense.In addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations.Numerical tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.展开更多
This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation $x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0)$ , where P n is a sequence ...This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation $x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0)$ , where P n is a sequence of nonnegative real numbers, and k n is a sequence of nonnegative integers with n ? k n →∞ as n → ∞.展开更多
The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the ine...The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the inertial manifolds of the nonautonomous retard parabolic equations are constructed by using the Lyapunov-Perron method.展开更多
This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous(cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets,Bohr/Levitan almost per...This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous(cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets,Bohr/Levitan almost periodic and almost automorphic motions, global attractors, and pinched and minimalsets is given. An application of our general results is given to scalar differential and difference equations.展开更多
It is shown that the nonautonomous discrete Toda equation and its Backlund transformation can be derived from the reduction of the hierarchy of the discrete KP equation and the discrete two-dimensional Toda equation. ...It is shown that the nonautonomous discrete Toda equation and its Backlund transformation can be derived from the reduction of the hierarchy of the discrete KP equation and the discrete two-dimensional Toda equation. Some explicit examples of the determinant solutions of the nonautonomous discrete Toda equation including the Askey-Wilson polynomial are presented. Finally we discuss the relationship between the nonautonomous discrete Toda system and the nonautonomous discrete Lotka- Volterra equation.展开更多
We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potenti...We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.展开更多
A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed an...A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficientsσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) on the collisions are graphically analyzed, whereσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. Whenσ1(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes.σ2(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter withσ3(x, t)=-1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).展开更多
基金Supported by the Natural Science Foundation of Guangdong Province（032469）
文摘By utilizing a fixed point theorem on cone, some new results on the existence ofpositive periodic solutions for nonautonomous differential equations with delay are derived.
基金the NNSFC(10771139 and 10771074)NSF of Wenzhou University(2007L024)NSF of Guangdong Province(004020077)
文摘This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.
文摘The long-time behaviour of a two-dimensional nonautonomous nonlinear Schrodinger equation is considered. The existence! of uniform attractor is proved and the upper bound of the uniform attractor's Housdorff dimension is given.
文摘In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is studied. A family of convergent approximate inertial manifolds for a class of evolution equations has been constructed when the spectral gap condition is satisfied.
文摘New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous difference equations is negative definite. In this paper, we consider the case that the Liapunov function is only positive definite and its variation is semi-negative definite. At these weaker conditions, we put forward a new asymptotical stability theorem of nonautonomous difference equations by adding to extra conditions on the variation. After that, in addition to the hypotheses of our new asymptotical stability theorem, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations provided that the Liapunov function has an indefinitely small upper bound. Example is given to verify our results in the last.
文摘The existence and the behavior of solutions to the differential-iterative equation arestudied without the restriction that f is monotone An error in and available paper is correctedhere.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175158)the Natural Science Foundation of Zhejiang Province of China(Grant No.LY12A04001)
文摘A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.
文摘The global attractivity of the delay difference equation Deltax(n) + a(n)x(n) + f(n, Sigma(s = -k)(0)(q) over bar (s), (n) x(s + n)) = 0, which includes the discrete type of many mathematical ecological equations, was discussed. The sufficient conditions that guarantee every solution to converge to zero are obtained. Many known results are improved and generated.
文摘In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomous evolution equations.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10875106 and 11175158)
文摘We present exact bright multi-soliton solutions of a generalized nonautonomous nonlinear Schroinger equation with time-and space-dependent distributed coefficients and an external potential which describes a pulse propagating in nonlinear media when its transverse and longitudinal directions are nonuniformly distributed.Such solutions exist in certain constraint conditions on the coefficients depicting dispersion,nonlinearity,and gain（loss）.Various shapes of bright solitons and interesting interactions between two solitons are observed.Physical applications of interest to the field and stability of the solitons are discussed.
基金National Natural Science Foundation of China Doctoral Program Foundation of the Ministry of Education of China
文摘Under the assumption that h(z) is strictly monotone the existence of solutions to a type of nonlinear differential-iterative equations in the form of x’(t) = g(x(t)) -h(x(x(t))) is discussed according to the behavior of the quasi-isoclinic curve C: x=h^-1 (g(t))
基金the National Natural Science Foundation of China(No.11671343).
文摘This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential equations.We prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square sense.In addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations.Numerical tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.
基金supported by Trans-Century Taining Programme Foundation for the Talents by the State Education Conmitteethe Doctoral Foundation of Ministry of Educatin of China(No.20020532014).
文摘This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation $x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0)$ , where P n is a sequence of nonnegative real numbers, and k n is a sequence of nonnegative integers with n ? k n →∞ as n → ∞.
文摘The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the inertial manifolds of the nonautonomous retard parabolic equations are constructed by using the Lyapunov-Perron method.
基金supported by the State Program of the Republic of Moldova “Multivalued Dynamical Systems, Singular Perturbations, Integral Operators and Non-Associative Algebraic Structures (Grant No. 20.80009.5007.25)”
文摘This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous(cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets,Bohr/Levitan almost periodic and almost automorphic motions, global attractors, and pinched and minimalsets is given. An application of our general results is given to scalar differential and difference equations.
基金supported in part by Grant-in-Aid for Scientific Research No. 18540214 from the Ministry of Education, Culture, Sports, Science, and Technology, Japan
文摘It is shown that the nonautonomous discrete Toda equation and its Backlund transformation can be derived from the reduction of the hierarchy of the discrete KP equation and the discrete two-dimensional Toda equation. Some explicit examples of the determinant solutions of the nonautonomous discrete Toda equation including the Askey-Wilson polynomial are presented. Finally we discuss the relationship between the nonautonomous discrete Toda system and the nonautonomous discrete Lotka- Volterra equation.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10875106 and 11175158
文摘We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.
基金Supported by the Fundamental Research Funds for the Central Universities under Grant No.2018MS132
文摘A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficientsσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) on the collisions are graphically analyzed, whereσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. Whenσ1(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes.σ2(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter withσ3(x, t)=-1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).
