Application of Fibonacci sine and cosine function to a nonlinear differential-difference equation

A new approach is constructed to obtain exact travelling wave solutions for a differential-difference equation by means of the property of the symmetrical Fibonacci sine and cosine function. As its illustration, some explicit and exact travelling wave solutions of Hybrid lattice, discretized mKdV lattice and modified Volterra lattice are obtained by computing the solutions of a lattice introduced by Wadati.     

Generalized Cauchy Matrix Approach for Lattice Boussinesq-Type Equations

The authors generalize the Cauchy matrix approach to get exact solutions to the lattice Boussinesq-type equations:lattice Boussinesq equation,lattice modified Boussinesq equation and lattice Schwarzian...     

Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Attractors for partly dissipative lattice dynamic systems in weighted spaces

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in weighted spaces. We first establish the dynamic systems on infinite lattice, and then prove the existence of the global attractor in weighted spaces by the asymptotic compactness of the solutions. It is shown that the global attractors contain traveling waves. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

The solutions of Toda lattice and Volterra lattice

This paper presents a method to directly construct explicit exact solutions to nonlinear differential-difference equations. One applies this approach to solve Volterra lattice and Toda lattice and obtain their some special solutions which contain soliton solutions and periodic solutions.     

准一维单原子晶格非线性振动方程的双周期解

利用行波变量代换和辅助椭圆方程法,求解了准一维单原子非线性晶格振动方程,得到了新的双周期波形式的椭圆函数解.在极限情形下,不仅可以还原为前人给出的扭结孤子解,同时还给出了一类新的类孤子解.     

一阶格点系统的不变测度与Liouville型方程

该文讨论一阶格点系统的解在相空间中的概率分布问题.作者先证明该格点系统的解算子生成的过程存在拉回吸引子,然后证明拉回吸引子上存在唯一的Borel不变概率测度,且该不变测度满足Liouville型方程.     

Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: a bilinear approach

The Kupershmidt five-field lattice is considered in this paper. By a dependent variable transformation, the Kupershmidt lattice is transformed into a bilinear form by the introduction of three auxiliary variables. We present a Bäcklund transformation and a nonlinear superposition formula for the Kupershmidt lattice. As an application of the results, soliton solutions are derived.     

Adomian decomposition method for nonlinear differential-difference equations

We extend Adomian decomposition method (ADM) to find the approximate solutions for the nonlinear differential-difference equations (NDDEs), such as the discretized mKdV lattice equation, the discretized nonlinear Schrödinger equation and the Toda lattice equation. By comparing the approximate solutions with the exact analytical solutions, we find the extend method for NDDEs is of good accuracy.     

Numerical solutions of two-dimensional Burgers’ equations by lattice Boltzmann method

In this paper, the two-dimensional Burgers’ equations with two variables are solved numerically by the lattice Boltzmann method. The lattice Bhatnagar–Gross–Krook model we used can recover the macroscopic equation with the second order accuracy. Numerical solutions for various values of Reynolds number, computational domain, initial and boundary conditions are calculated and validated against exact solutions or other published results. It is concluded that the proposed method performs well.     

A discrete generalization of the extended simplest equation method

We modified the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences.     

Integrable nonholonomic deformation of modified Volterra lattice equation

The modified Volterra lattice equation with nonholonomic constrain has been considered in this paper. The integrability of the deformed model has been demonstrated by providing a Lax pair. Applying the gauge transformation to the Lax pair, we establish Darboux transformation theorem for the nonholonomic deformation equation. Some analytic solutions of the system are obtained via the one-fold and two-fold Darboux transformations. The deformation on explicit solutions exhibits different curvy profiles and propagation trajectories that were not found in modified Volterra lattice equation.     

Solution of a nonlinear Schr?dinger equation in the form of two-phase freak waves

We construct a family of two-gap solutions of the focusing nonlinear Schr?dinger equation and derive a condition under which the solutions behave as the so-called freak waves located at the nodes of a two-dimensional lattice. We also study how the lattice parameters depend on the parameters of the spectral curve.     

KGS格点系统的全局吸引子

考虑了对应于Klein-Gordon-Schrdinger方程的格点系统(KGS格点系统)的解的长时间行为．首先通过引入一个加权范数与采用解的“切尾”法,证明了全局吸引子的存在性．在此基础上,采用元素分解法与多面体的球覆盖性质, 得到了此吸引子的Kolmogorov δ-熵的上界的一个估计．最后,我们用有限维的常微分方程的全局吸引子逼近它．     

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.     

Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.     

Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.     

Explicit Solutions to an Integrable Lattice

In this paper, we aim to construct the Darboux transformation and explicit solutions for an integrable lattice introduced by Suris. Analysis of properties of the solutions shows that the obtained explicit solutions for this discrete integrable system possess new dynamical characters which are different from the ones of continuous integrable systems.     

Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise

In this paper, we study the long-term asymptotic behaviour of solutions to the stochastic Zakharov lattice equation with multiplicative white noise. We first transfer the stochastic lattice equation into a random lattice equation and prove the existence and uniqueness of solutions which generate a random dynamical system. Then we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Finally we establish the upper semi-continuity of random attractor to the global attractor of the limiting system as the coefficients of the white noise terms tend to zero.