廣義 Riccati展開法及其在foam drainage方程中的應用

費琪

(1.西北大學數(shù)學系,陜西 西安 710127;2.西北大學非線性中心,陜西 西安 710069)

廣義 Riccati展開法及其在foam drainage方程中的應用

費琪1,2

(1.西北大學數(shù)學系,陜西 西安 710127;2.西北大學非線性中心,陜西 西安 710069)

應用雙曲函數(shù)法結(jié)合Riccati方程,求得foam drainage方程的精確解.通過這種方法可以得到此方程的新的孤立波解與周期解,并且此方法可以用來求解其它許多的非線性演化方程.

雙曲函數(shù)法結(jié)合Riccati方程;行波解;foam drainage方程

1 引言

非線性現(xiàn)象在各科學領(lǐng)域與工程領(lǐng)域普遍存在,近些年來,解方程的方法層出不窮,如逆射法[1,2],雙曲函數(shù)法[3],sine-cosine法[4],tanh-sech法[5],F展開法[6],G′/G展開法[7]等. Mal fi et首先用tanh方法解非線性偏微分方程,后來此方法被廣泛應用,Wazwas把這種方法推廣為tanh-coth方法.本文運用tanh-coth法結(jié)合Riccati方程求解foam drainage方程新的行波解,同時體現(xiàn)出這種方法解非線性方程非常直觀有效.

泡沫出現(xiàn)在日常生活和工業(yè)中,foam drainage方程描述了在重力的作用下,泡沫垂直密度的變化情況.這個模型由Verbist和Weaire在1996年提出,它的解在食品生產(chǎn)、化學工業(yè)、消防、材料物理等領(lǐng)域中有很重要的作用,因此研究foam drainage方程在理論和實踐中有著

很重要的意義.

2 雙曲函數(shù)法結(jié)合Riccati方程基本思想及求解步驟

3 foam drainage方程

4 結(jié)論

本文在Maple軟件運算功能的幫助下,運用了雙曲函數(shù)法結(jié)合廣義Riccati方程成功求得了foam drainage方程新的行波解,這樣豐富了方程解的結(jié)果,有助于對方程所描述的物理現(xiàn)象有進一步的了解和研究.

[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.

國家自然科學基金(10671156).

費琪(1985-),碩士生,研究方向：偏微分方程.

2010 MSC:35Q58