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.     

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.     

A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves

A new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations. The solutions obtained in this paper include soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.     

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.     

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.     

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.     

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.     

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.     

An algebraic method exactly solving two high-dimensional nonlinear evolution equations

An algebraic method is applied to construct soliton solutions, doubly periodic solutions and a range of other solutions of physical interest for two high-dimensional nonlinear evolution equations. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solutions according to some parameters.     

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.     

Classification of optical wave solutions to the nonlinearly dispersive Schrödinger equation

Different kinds of optical wave solutions to the nonlinearly dispersive Schrödinger equation are given according to different parameters’ regions. Those solutions include looped wave solutions, cusped wave solutions, peaked wave solutions, compacted wave solutions. The looped and cusped forms have not been reported in the literature regarding to the study of the nonlinear Schrödinger equation. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.     

Rational Solutions to the KdV Equation in Terms of Particular Polynomials

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n$, and we call these solutions, solutions of the order $n$. Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order $10$.     

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.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.     

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。     

A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation

A new algebraic method is devised to obtain a series of exact solutions for general nonlinear equations. Compared with the most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters. For illustration, we apply the method to solve a new two-dimensional perturbed KdV equation and successfully construct the various kind of exact solutions including line soliton solutions, rational solutions, triangular periodic solutions, Jacobi, and Weierstrass doubly periodic solutions.     

带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限

研究了带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限.由于非齐次项的影响,带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解不再是自相似的.当压力和磁感强度同时消失时,它的解会收敛到零压流输运方程组的Riemann解,解中会出现δ-激波和真空现象.同时研究还得到了仅当磁感强度消失时,它的解会收敛到非齐次广义Chaplygin气体Euler方程组的Riemann解,并且解中只出现δ-激波.     

Hierarchies of exact solutions for the discrete third Painlevé equation

In this paper, we construct hierarchies of rational solutions of the discrete third Painlevé equation (d-PIII) by applying Schlesinger transformations to simple initial solutions. We show how these solutions reduce in the continuous limit to the hierarchies of rational solutions of the third Painlevé equation (PIII). We also study the solutions of d-PIII which are expressed in terms of discrete Bessel functions and show that these solutions reduce in the continuous limit the hierarchies of special function solutions of PIII.     

Some exact solutions of KdV equation with variable coefficients

In this paper, the extended mapping transformation method is used to obtain some new exact solutions of a variable-coefficient KdV equation arising in arterial mechanics. The obtained solutions include soliton solutions, periodic solutions and rational solutions.