On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparis...On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.展开更多
In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic ...In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schro¨dinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.展开更多
Adomian decomposition method is applied to find the analytical and numerical solutions for the discretizedmKdV equation.A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation.T...Adomian decomposition method is applied to find the analytical and numerical solutions for the discretizedmKdV equation.A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation.The procedure presented here can be used to solve other differential-difference equations.展开更多
The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using l...The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction.展开更多
In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we pr...In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.展开更多
We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schrodinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of...We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schrodinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t-1/2.展开更多
We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real val...We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.展开更多
文摘On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.61072147,11071159)the Natural Science Foundation of Shanghai Municipality (Grant No.09ZR1410800)+1 种基金the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No.KLMM0806)the Shanghai Leading Academic Discipline Project (Grant Nos.J50101, S30104)
文摘In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schro¨dinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.
基金The project supported by National Natural Science Foundation of China under Grant No.10672147the Natural Science Foundation of Zhejiang Province under Grant No.Y605312
文摘Adomian decomposition method is applied to find the analytical and numerical solutions for the discretizedmKdV equation.A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation.The procedure presented here can be used to solve other differential-difference equations.
基金Supported by National Natural Science Foundation of China under Grant No.11271246Natural Science Research Project of Henan Education Department under Grant No.2011B110024+1 种基金Research Fund for Luoyang Normal University under Grant No.qnjj-2009-02for Henan Polytechnic University under Grant No.Q2012-30A
文摘The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction.
基金Supported by Science and technology plan foundation of Guangzhou(No.201607010218)by Public Research&Capacity-Building Project of Guangdong(No.2015A070704059).
文摘In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
文摘We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schrodinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t-1/2.
基金Supported by NSFC(Grant No.11571370)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20120162110021)of China
文摘We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.
