非线性演化方程的Painlevé分析
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摘要
Painlevé分析既可以用来判断非线性演化方程的可积性,又可以用来求出其精确解,故被广泛应用到非线性系统的研究中。以Burgers方程和KdV方程为例,详细分析了非线性演化方程Painlevé性质的两种重要检验方法——WTC方法和Kruskal简化法。相比WTC方法,Kruskal简化法可以更为快速地判定非线性演化方程的Painlevé可积性。两种方法为寻找新的Painlevé可积系统提供了重要途径。
Painleve analysis can be used not only to judge the integrability of nonlinear evolution equations, but also to obtain their exact solutions. It is widely applied to the study of nonlinear systems.Burgers equation and KdV equation are taken as examples to carry out analysis in detail for two important test methods, namely, WTC method and Kruskal simplification method, for the Painleve property of nonlinear evolution equation. Compared to WTC method, one can judge the Painleve integrability of nonlinear evolution equations more rapidly using Kruskal simplification method. The two methods provide an important way to find new Painleve integrable systems.
Painleve analysis can be used not only to judge the integrability of nonlinear evolution equations, but also to obtain their exact solutions. It is widely applied to the study of nonlinear systems.Burgers equation and KdV equation are taken as examples to carry out analysis in detail for two important test methods, namely, WTC method and Kruskal simplification method, for the Painleve property of nonlinear evolution equation. Compared to WTC method, one can judge the Painleve integrability of nonlinear evolution equations more rapidly using Kruskal simplification method. The two methods provide an important way to find new Painleve integrable systems.
引文
[1] Weiss J,Tabor M,Carnevale G. The Painleve property for partial differential equations[J]. Journal of Mathematical Physics, 1983, 24(3):522-526.
[2] Weiss J. The Painleve property for partial differential equation.Ⅱ:Backlund transformation,Lax pairs, and the Schwarzian derivative[J]. Journal of Mathematical Physics,1983, 24(6):1405-1413.
[3]尤月东.一类非线性偏微分方程的Painleve分析[D].呼和浩特:内蒙古工业大学,2017.
[4]徐桂琼.非线性演化方程的精确解与可积性及其符号计算研究[D].上海:华东师范大学,2004.
[5] Jimbo M, Kruskal M. D, Miwa T. Painleve test for the self-dual Yang-Mills equation[J]. Physics Letters A, 1982,92(2):59-60.
[6]李翊神.孤子与可积系统[M].上海:上海科技教育出版社,1999.
[2] Weiss J. The Painleve property for partial differential equation.Ⅱ:Backlund transformation,Lax pairs, and the Schwarzian derivative[J]. Journal of Mathematical Physics,1983, 24(6):1405-1413.
[3]尤月东.一类非线性偏微分方程的Painleve分析[D].呼和浩特:内蒙古工业大学,2017.
[4]徐桂琼.非线性演化方程的精确解与可积性及其符号计算研究[D].上海:华东师范大学,2004.
[5] Jimbo M, Kruskal M. D, Miwa T. Painleve test for the self-dual Yang-Mills equation[J]. Physics Letters A, 1982,92(2):59-60.
[6]李翊神.孤子与可积系统[M].上海:上海科技教育出版社,1999.