基于雙線性算子計算薛定諤方程組的精確解

趙敬宜,李壹宏

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

基于雙線性算子計算薛定諤方程組的精確解

趙敬宜1,2,李壹宏1,2

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

通過引進雙線性算子,用合適的變換將已知方程組轉(zhuǎn)化為雙線性方程,并用擾動法求得方程組的單孤子和雙孤子解.其結(jié)果有助于對數(shù)學方程的理解和在物理中的應用.

雙線性算子;薛定諤方程組;孤子解

DO I：10.3969/j.issn.1008-5513.2013.04.008

1 引言

在廣泛的流體力學問題中都會遇到Davey-Stewartson方程組(簡稱為DS方程組).其形式可寫為[1]:

其中u(x,y,t)為復函數(shù),v(x,y,t)為實函數(shù),γ,β為非線性系數(shù),α為相互作用系數(shù).而隨著非線性數(shù)學物理方程的廣泛應用,對于求此類方程的解已有許多方法[2-4],如逆反射法、Painlev′e展開法、B¨acklund變換法、Jacobian橢圓函數(shù)法等等,此方程也可看作是1+1維非線性薛定諤方程即NLS方程向2+1維的拓廣[3].本文主要應用Hirota雙線性法[5]求NLS方程組的精確孤子解,其形式為[6]:

式中β,γ,δ為任意實常數(shù).

2 求解非線性方程組的H irota直接法

3 N LS方程組的單孤子解

4 N LS方程組的雙孤子解

[1]Davey A,Stewartson K.On three-dimensional packets of surfacewaves[J].Proceedings of the Royal Society of London.A.M athem atical and Physical Sciences,1974,338(1613):101-110.

[2]劉式適,劉式達.物理學中的非線性問題[M].北京：北京大學出版社,2000.

[3]李翊神.孤立子可積系統(tǒng)[M].上海:上海科技教育出版社,1999.

[4]A rkadiev V A,Pogrebkov A K,Polivanov M C.Inverse scattering transform m ethod and soliton solutions for Davey-Stewartson II equation[J].Physica D:Non linear Phenom ena,1989,36(1):189-197.

[5]Hirota R.The Direct M ethod in Soliton Theory[M].Cambridge:Cambridge University Press,2004.

[6]錢賢民,張葉.雙線性算子和非線性方程的雙線性形式[J].紹興文理學院學報:自然科學版,2002,22(10):10-15. [7]高斌,劉式達,劉式適.Davey-Stewartson方程組的包絡(luò)周期解和孤立波解[J].物理學報,2009,58(4):2155-2158.

Exact solu tions of a coup led Sch r¨od inger equations based on
b ilinear operator

Zhao Jingyi1,2,Li Yihong1,2
(1.Department of M athematics,Northwest University,Xi′an 710127,China; 2.Center for Nonlinear Studies,Northwest University,Xi′an 710069,China)

The non linear Schr¨odinger equation is transformed and expressed by using bilinear operator,and the soliton solutions are obtained through the perturbation method.M eanwhile,the result helps us to learn more aboutmathematics equation and the equation applied in physics.

bilinear operator,a coup led Sch r¨od inger equations,soliton solution

O 175.2

A

1008-5513(2013)04-0382-05

2012-12-18.

趙敬宜(1989-),碩士生,研究方向:非線性偏微分方程.

2010 M SC:35J15