一类广义Canard系统的近似解析解

利用变分迭代理论研究了一类广义Canard系统. 首先引入一组泛函,然后构造了原方程解的迭代关系式,最后得到了问题鸭解的近似解析式. 关键词： 鸭解 解析解 变分迭代     

(3+1)维Burgers扰动系统孤波的解法

研究了在物理模型中的一类扰动高维非线性Burgers系统. 利用经过改进的广义变分迭代方法, 构造了相应迭代关系式. 得到了扰动系统的孤波近似解.     

广义(3+1)维非线性Burgers系统孤波级数解

研究了一类广义(3+1)维非线性Burgers系统.首先,利用同伦映射方法构造了相应的映射关系式.其次,利用迭代方法得到了扰动系统的一个孤波非行波的级数解.     

一类广义Sine-Gordon扰动方程的解析解

利用同伦映射方法研究了一类广义Sine-Gordon方程. 首先引入一个同伦变换. 然后构造了原方程解的迭代关系式. 最后得到了问题的解析解. 关键词： 孤子 扰动 同伦映射     

椭圆旋轮线

一条光滑曲线在另一条光滑曲线上无滑动滚动,滚动曲线上一点的轨迹就是广义旋轮线.利用弧长坐标,给出了广义旋轮线的轨迹方程.对于椭圆,画出了椭圆旋轮线轨迹.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov方程作为一类重要的非线性方程有着许多广泛的应用前景,基于Hamilton空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值解法,讨论了利用Preissmann方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov 方程作为一类重要的非线性方程有着许广泛的应 用前景,基于Hamilton 空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值 解法,讨论了利用Preissmann 方法构造离散多辛格式的途径, 并构造了一种典型的半隐 式的多辛格式, 该格式满足多辛守恒律、局部能量守恒律. 数值算例结果表明该多辛离 散格式具有较好的长时间数值稳定性.     

非线性广义Landau-Ginzburg-Higgs方程孤子解的变分迭代解法

利用扰动方法研究了一类非线性广义Landau-Ginzburg-Higgs方程.首先引入一个泛函，计算它的变分，求得Lagrange乘子.然后构造了原方程解的迭代关系式，得到了近似解. 关键词： 孤子 扰动 变分迭代     

一类广义Birkhoff系统的广义正则变换

研究一类广义Birkhoff系统的广义正则变换.建立这类广义Birkhoff系统的运动微分方程,得到了该系统的广义正则变换以及保持广义正则变换的条件.最后,举例说明结果的应用.     

离散时间切换广义系统的一致有限时间稳定性

针对一类离散时间切换广义系统, 研究其一致有限时间稳定性问题. 首先, 把广义系统的有限时间稳定性概念推广到离散切换广义系统; 然后, 利用Lyapunov-like函数方法, 给出了离散切换广义系统在任意给定的切换规则下是正则、 因果的, 且有限时间有界和有限时间稳定的充分条件, 同时给出了保证离散切换广义系统一致有限时间稳定的状态反馈控制器的具体设计方法. 仿真算例结果说明了该控制方法的有效性.     

Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system

We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node-type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualization of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to find and visualize canard solutions as the intersection curves of the attracting and repelling slow manifolds.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Exact Solutions to Maccari's System

Based on the generalized Riccati relation, an algebraic method to construct a series of exact solutions to nonlinear evolution equations is proposed. Being concise and straightforward, the method is applied to Maccari's system, and some exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Exact Solutions to Maccari's System

Based on the generalized Riccati relation, an algebraic method to construct a series of exact solutions to nonlinear evolution equations is proposed. Being concise and straightforward, the method is applied to Maccari's system, and some exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Feedback control of canards

We present a control mechanism for tuning a fast-slow dynamical system undergoing a supercritical Hopf bifurcation to be in the canard regime, the tiny parameter window between small and large periodic behavior. Our control strategy uses continuous feedback control via a slow control variable to cause the system to drift on average toward canard orbits. We apply this to tune the FitzHugh-Nagumo model to produce maximal canard orbits. When the controller is improperly configured, periodic or chaotic mixed-mode oscillations are found. We also investigate the effects of noise on this control mechanism. Finally, we demonstrate that a sensor tuned in this way to operate near the canard regime can detect tiny changes in system parameters.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

构造非线性发展方程无穷序列复合型精确解的一种方法

为了获得非线性发展方程新的无穷序列复合型精确解,给出了Riccati方程的Bäcklund变换和解的非线性叠加公式,符号计算系统Mathematica的帮助下,以广义Boussinesq方程为应用实例,获得了无穷序列复合型精确解.这里包括双曲函数、三角函数与有理函数复合解、双曲函数与三角函数复合解等几种新的无穷序列复合型精确解.该方法在构造非线性发展方程无穷序列复合型精确解方面具有普遍意义. 关键词： 非线性发展方程 非线性叠加公式 Riccati方程 无穷序列精确解     

Explicit solutions of two nonlinear dispersive equations with variable coefficients

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact solutions for a generalized Camassa-Holm equation and a nonlinear dispersive equation with variable coefficients. It is shown that the variable coefficients of the derivative terms in the equations cause the qualitative change in the physical structures of the solutions.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

Spatiotemporal Self-Similar Solutions of the Generalized (3+1)-dimensional Nonlinear Schrödinger Equation with Polynomial Nonlinearity of Arbitrary Order

We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrödinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quintic nonlinear Schrödinger equation with distributed dispersion and distributed linear gain, including bright similariton solution, fractional and combined Jacobian elliptic function solutions. Moreover, we discuss self-similar evolutional dynamic behaviors of these solutions in the dispersion decreasing fiber and the periodic distributed amplification system.