Periodic Wave Solutions and Their Limits for the Modified Kd V–KP Equations

In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified Kd V–KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.     

关于运输问题最优解的进一步讨论

首先给出了运输问题最优解的相关概念,将最优解扩展到广义范畴,提出狭义多重最优解和广义多重最优解的概念及其区别.然后给出了惟一最优解、多重最优解、广义有限多重最优解、广义无限多重最优解的判定定理及其证明过程.最后推导出了狭义有限多重最优解个数下限和广义有限多重最优解个数上限的计算公式,并举例验证了结论的正确性.     

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

Several classes of exact solutions to the 1 + 1 Born–Infeld equation

We obtain closed-form exact solutions to the 1 + 1 Born–Infeld equation arising in nonlinear electrodynamics. In particular, we obtain general traveling wave solutions of one wave variable, solutions of two wave variables, similarity solutions, multiplicatively separable solutions, and additively separable solutions. Then, putting the Born–Infeld model into correspondence with the minimal surface equation using a Wick rotation, we are able to construct complex helicoid solutions, transformed catenoid solutions, and complex analogues of Scherk’s first and second surfaces. Some of the obtained solutions are new, whereas others are generalizations of solutions in the literature. These exact solutions demonstrate the fact that solutions to the Born–Infeld model can exhibit a variety of behaviors. Exploiting the integrability of the Born–Infeld equation, the solutions are constructed elegantly, without the need for complicated analytical algorithms.     

小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

Symbolic computation of exact solutions for a nonlinear evolution equation

In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

纤维混凝土中应力传递机制的三维弹性理论分析

对纤维混凝土中应力传递机制问题作了分析研究,将应力和位移看作一组“初等解”和一组“修正解”的叠加,“初等解”即为一般的二维理论所得的解答,而“修正解”应用拉甫（Love)位移函数法求得,计算实例表明,解的收敛性良好,将三维弹性理论解与剪滞法的解比较后,可见两种解存在较明显的差别。     

Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters北大核心CSCD

利用多项式完全判别系统法求得非线性光学中带参数时空分数阶Fokas-Lenells方程在一般情况下的精确解,包括有理函数解、周期解、孤波解、Jacobi椭圆函数解和双曲函数解等,绘制了精确解的相关图像,并由此分析了参数对解的结构的影响。     

The extended hyperbolic functions method and new exact solutions to the Zakharov equations

The multiple exact solutions for the nonlinear evolution equations describing the interaction of laser–plasma are developed. The extended hyperbolic function method are employed to reveal these new solutions. The solutions include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the solitary wave solutions of a compound of the bell-type and the kink-type for n and E, the singular traveling wave solutions, periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. In addition to re-deriving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method.     

New double Wronskian solutions of the AKNS equation

Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction.     

F展开法结合指数函数法求解Zhiber-Shabat 方程的精确解

利用F展开法与指数函数法相结合的方法,在相关文献的基础上,重新研究了Zhiber-Shabat方程,获得了许多与现有文献中解的表达式不相同的各种精确解.这些解同样具有孤立波解,纽子波解和周期波解的各种动力学特征.从而丰富了相关文献中关于Zhiber-Shabat波方程的孤立子解和周期解的种类.     

New exact solutions to the (2 + 1)-dimensional Ito equation: Extended homoclinic test technique

Exact soliton solutions to the (2 + 1)-dimensional Ito equation are studied based on the idea of extended homoclinic test and bilinear method. Some explicit solutions, such as triangle function solutions, soliton solutions, doubly-periodic wave solutions and periodic solitary wave solutions, are obtained. It shows that the (2 + 1)-dimensional Ito equation has richer solutions. Besides, the elastic interactions of the solutions and their corresponding physical meaning are discussed.     

Some new and general solutions to the compound KdV-Burgers system with nonlinear terms of any order

In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained.     

KK方程和改进的Boussinesq方程的新精确解

应用改进的Fan's代数方法,得到了KK方程和改进的Boussinesq方程的一系列新精确解,包括孤立波解、类孤立波解、纽结波解、奇异纽结波解和三角函数周期解.     

Travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation

The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also give...     

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

Explicit Traveling Wave Solutions and Their Dynamical Behaviors for the Coupled Higgs Field Equation

In this paper, we focus on the traveling wave solutions of the coupled Higgs field equation from the perspective of dynamical systems. Through the phase portraits, in addition to periodic wave solutions and solitary wave solutions, we also obtain explicit periodic singular wave solutions, singular wave solutions and kink wave solutions, which were not found in the previous works. The dynamical behavior of these solutions and their internal relations are revealed through asymptotic analysis. The results will help supplement the study of field equation.