广义 Riccati展开法及其在foam drainage方程中的应用

费琪

(1.西北大学数学系,陕西 西安 710127;2.西北大学非线性中心,陕西 西安 710069)

广义 Riccati展开法及其在foam drainage方程中的应用

费琪1,2

(1.西北大学数学系,陕西 西安 710127;2.西北大学非线性中心,陕西 西安 710069)

应用双曲函数法结合Riccati方程,求得foam drainage方程的精确解.通过这种方法可以得到此方程的新的孤立波解与周期解,并且此方法可以用来求解其它许多的非线性演化方程.

双曲函数法结合Riccati方程;行波解;foam drainage方程

1 引言

非线性现象在各科学领域与工程领域普遍存在,近些年来,解方程的方法层出不穷,如逆射法[1,2],双曲函数法[3],sine-cosine法[4],tanh-sech法[5],F展开法[6],G′/G展开法[7]等. Mal fi et首先用tanh方法解非线性偏微分方程,后来此方法被广泛应用,Wazwas把这种方法推广为tanh-coth方法.本文运用tanh-coth法结合Riccati方程求解foam drainage方程新的行波解,同时体现出这种方法解非线性方程非常直观有效.

泡沫出现在日常生活和工业中,foam drainage方程描述了在重力的作用下,泡沫垂直密度的变化情况.这个模型由Verbist和Weaire在1996年提出,它的解在食品生产、化学工业、消防、材料物理等领域中有很重要的作用,因此研究foam drainage方程在理论和实践中有着

很重要的意义.

2 双曲函数法结合Riccati方程基本思想及求解步骤

3 foam drainage方程

4 结论

本文在Maple软件运算功能的帮助下,运用了双曲函数法结合广义Riccati方程成功求得了foam drainage方程新的行波解,这样丰富了方程解的结果,有助于对方程所描述的物理现象有进一步的了解和研究.

[1]Ablowitz M J,Segur H.Solitons and Inverse Scattering Transform[M].Philadelphia:SIAM,1981.

[2]Vakhnenko V O,Parkes E J,Morrison A J.A Backlund transformation and the inverse scattering transform method for the generalised Vakhnenko equation[J].Chaos,Solitons and Fractals,2003,17(4):683-92.

[3]Fan Engui．Extended tanh-function method and its applications to nonlinear equations[J]．Physics Letters A,2000,277:212-218．

[4]Bekir A.New solitons and periodic wave solutions for some nonlinear physical models by using sine-cosine method[J].Physica Scripta,2008,77:501-504.

[5]Mal fl iet W.Solitary wave solutions of non-linear wave equations[J].Am.J.Phys.,1992,60:650-654.

[6]Abdou M A.Further improved F-expansion and new exact solutions for nonlinear evolution equations[J]. Nonlinear Dynam.,2008,52(3):277-288.

[7]Bekir A.Application of the(G′/G)-expansion method for nonlinear evolution equations[J].Phys.Lett.A, 2008,372:340-356.

Generalized Riccati equation expansion method and its application to the foam drainage equation

Fei Qi
(1.Department of Mathematics,Northwest University,Xi′an 710127,China;
2.Center for Nonlinear Studies,Northwest University,Xi′an 710069,China)

In this paper,the tanh-coth method combined with the Riccati equation is applied to the analysis of the foam drainage equation.We can gain new solitary wave solutions and periodic solutions.It is also a promising method to solve other nonlinear evolution equations.

the tanh-coth method combined with the Riccati equation,travelling wave, the foam drainage equation

O175.2

A

1008-5513(2012)01-0109-04

2011-02-12.

国家自然科学基金(10671156).

费琪(1985-),硕士生,研究方向：偏微分方程.

2010 MSC:35Q58