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In this article, we introduce some results with respect to the integrality and exact solutions of some 2nd order algebraic DEs. We obtain the sufficient and necessary conditions of integrable and the general meromorphic solutions of these equations by the complex method, which improves the corresponding results obtained by many authors. Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics. 相似文献
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We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations. 相似文献
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Zifeng Huang Liming Zhang Qiuhui Chen Wenjun Yuan 《Mathematical Methods in the Applied Sciences》2014,37(10):1553-1560
In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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The exponential function rational expansion method and exact solutions to nonlinear lattice equations system 总被引:1,自引:0,他引:1
Hua Xin 《Applied mathematics and computation》2010,217(4):1561-1565
We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem. 相似文献
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Wenjun Yuan Fanning Meng Jianming Lin Yonghong Wu 《Mathematical Methods in the Applied Sciences》2016,39(8):2083-2092
Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Fanning Meng Liming Zhang Yonghong Wu Wenjun Yuan 《Mathematical Methods in the Applied Sciences》2015,38(17):3678-3688
In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg‐de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w2r,2(z) and simply periodic solutions w1s,2(z),w2s,1(z) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations. 相似文献
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本文讨论了一类复微分方程组的亚纯解的分量的Nevanlinna特征估计,利用亚纯函数Nevanlinna值分布理论,得到了一个有关方程组亚纯解的分量的Nevanlinna特征估计定理.该结果推广了高凌云等人的一些结果. 相似文献
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Zhengxin Zhou 《Applied mathematics and computation》2012,218(9):5671-5681
In this article, we use the new method of reflecting function to study the behavior of solutions of nonlinear time-vary differential equations, and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional. We applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center. 相似文献
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给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解. 相似文献
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mKdV方程和mKP方程组的新的精确孤立波解 总被引:2,自引:0,他引:2
用三角函数假设法和一种新辅助方程的解构造mK dV方程和mKP方程组的精确孤立波解.这种方法也可用于寻找其它非线性发展方程的新的孤立波解. 相似文献
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本运用Liapunov函数方法,研究了一类四阶非线性微分方程的周期解,得到了存在唯一渐近稳定的周期解的充分条件。 相似文献
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构造非线性差分方程精确解的一种方法 总被引:1,自引:0,他引:1
套格图案 《高校应用数学学报(A辑)》2008,23(3)
在齐次平衡法、试探函数法的基础上,给出指数函数所组成的两种试探函数法,并借助符号计算系统Mathematica构造了Hybrid-Lattice系统、mKdV差分微分方程、Ablowitz-Ladik.Lattice6系统等非线性离散系统的新的精确孤波解. 相似文献
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一类求行波解的线性方法 总被引:2,自引:0,他引:2
基于齐次平衡法和李志斌的 tanh函数法 ,本文得到一类简单有效的求解非线性发展方程的线性方法 .这类方法利用非线性发展方程孤立波的局部性特点 ,适当地选取函数 f 和 g,将孤波表示为 f,g的多项式 ,从而将非线性发展方程求解问题转化为非线性代数方程组的求解问题 ,再利用吴消元法求解方程组从而得到非线性发展方程的行波解 相似文献
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Jing Wang 《Applied mathematics and computation》2010,217(4):1652-1657
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. 相似文献
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主要利用Jacobi椭圆函数所满足的方程并用其解代替Jacobi椭圆函数以求非线性偏微分方程的周期解,并举例说明该方法的应用. 相似文献
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本文运用 L iapunov函数方法 ,研究了一类三阶非线性微分方程的周期解 ,得到了存在唯一渐近稳定的周期解的充分条件 相似文献
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ON THE EQUIVALENCE OF THE ABEL EQUATION 总被引:2,自引:0,他引:2
This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations. 相似文献