一类四阶偏微分方程的李对称分析、B?cklund变换及其精确解
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摘要
利用齐次平衡法获得了一类四阶偏微分方程的B?cklund变换,进而得到方程的几组精确解;然后运用李对称分析方法,获得该方程的向量场,利用相似变换,把难于求解的非线性偏微分方程转化为易于求解的常微分方程,并通过求解所得到的约化方程,结合幂级数展开法,得到原方程的一系列精确解.
This paper investigates a class of variable coefficient partial differential equations. By using the homogeneous balance method, the Backlund transformation is obtained, which leads to exact solutions for the equation. By applying Lie symmetry analysis, the symmetries and vector field of the equation are obtained. Then, by means of a similarity transformation,the partial differential equations are reduced to ordinary differential equations. Solving the reduced equations, we investigate the exact solutions to the equations concisely using the power series expansion method.
This paper investigates a class of variable coefficient partial differential equations. By using the homogeneous balance method, the Backlund transformation is obtained, which leads to exact solutions for the equation. By applying Lie symmetry analysis, the symmetries and vector field of the equation are obtained. Then, by means of a similarity transformation,the partial differential equations are reduced to ordinary differential equations. Solving the reduced equations, we investigate the exact solutions to the equations concisely using the power series expansion method.
引文
[1]李宁,刘希强.Broer-Kau-Kupershmidt方程组的对称,约化和精确解[J].物理学报,2013, 62(16):160203-1-160203-6.
[2]楼森岳.推广的Painlevé展开及KdV方程的非标准截断解[J).物理学报,1998, 47(12):1937-1945.
[3]张金华,丁玉敏.F展开法结合指数函数法求解Zhiber-Shabat方程的精确解[J].纯粹数学与应用数学,2011, 27(2):163-169.
[4]刘式适,傅遵涛,刘式达,等.Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用[J].物理学报,2001, 50(11):2068-2073.
[5]范恩贵,张鸿庆.齐次平衡法若干新的应用[J].数学物理学报,1999, 19(3):286-292.
[6]HIROTA R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[J]. Physical Review Letters, 1971, 27(18):1456-1458.
[7]斯琴.同伦摄动法求KdV方程和Burgers方程的近似解[J].内蒙古师范大学学报(自然科学版),2008, 37(3):381-384.
[8] SAHOO S,GARAI G,RAY S S. Lie symmetry analysis for similarity reduction and exact solutions of modified KdV-Zakharov-Kuznetsov equation[J]. Nonlinear Dynamics, 2017, 87(3):1995-2000.
[9] SAHOO S, RAY S S. Lie symmetry analysis and exact solutions of(3+1)dimensional Yu-Toda-Sasa-Fukuyama equation in mathematical physics[J]. Computers&Mathematics with Applications, 2017, 73(2):253-260.
[10]刘汉泽.基于李对称分析的偏微分方程精确解的研究[D].昆明:昆明理工大学,2009.
[11] ZAYED E M E, ALURRFI K A E. The B(a|¨)cklund transformation of the Riccati equation and its applications to the generalized KdV-mKdV equation with any-order nonlinear terms[J]. PanAmerican Mathematical Journal,2016, 26(2):50-62.
[12]曾昕,张鸿庆.(2+1)维Boussinesq方程的Backlund变换与精确解[J].物理学报,2005, 54(4):1476-1480.
[13] MINGLIANG W. Exact solutions for a compound KdV-Burgers equation[J]. Phys Letters A, 1996, 213:279-287.
[14] YAO Q. Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem[J]. Nonlinear Analysis:Theory, Methods&Applications, 2005, 63(2):237-246.
[15] ROGERS C, SHADWICK W F. B(a|¨)cklund Transformations and Their Applications[M]. New York:Academic Press,1982.
[2]楼森岳.推广的Painlevé展开及KdV方程的非标准截断解[J).物理学报,1998, 47(12):1937-1945.
[3]张金华,丁玉敏.F展开法结合指数函数法求解Zhiber-Shabat方程的精确解[J].纯粹数学与应用数学,2011, 27(2):163-169.
[4]刘式适,傅遵涛,刘式达,等.Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用[J].物理学报,2001, 50(11):2068-2073.
[5]范恩贵,张鸿庆.齐次平衡法若干新的应用[J].数学物理学报,1999, 19(3):286-292.
[6]HIROTA R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[J]. Physical Review Letters, 1971, 27(18):1456-1458.
[7]斯琴.同伦摄动法求KdV方程和Burgers方程的近似解[J].内蒙古师范大学学报(自然科学版),2008, 37(3):381-384.
[8] SAHOO S,GARAI G,RAY S S. Lie symmetry analysis for similarity reduction and exact solutions of modified KdV-Zakharov-Kuznetsov equation[J]. Nonlinear Dynamics, 2017, 87(3):1995-2000.
[9] SAHOO S, RAY S S. Lie symmetry analysis and exact solutions of(3+1)dimensional Yu-Toda-Sasa-Fukuyama equation in mathematical physics[J]. Computers&Mathematics with Applications, 2017, 73(2):253-260.
[10]刘汉泽.基于李对称分析的偏微分方程精确解的研究[D].昆明:昆明理工大学,2009.
[11] ZAYED E M E, ALURRFI K A E. The B(a|¨)cklund transformation of the Riccati equation and its applications to the generalized KdV-mKdV equation with any-order nonlinear terms[J]. PanAmerican Mathematical Journal,2016, 26(2):50-62.
[12]曾昕,张鸿庆.(2+1)维Boussinesq方程的Backlund变换与精确解[J].物理学报,2005, 54(4):1476-1480.
[13] MINGLIANG W. Exact solutions for a compound KdV-Burgers equation[J]. Phys Letters A, 1996, 213:279-287.
[14] YAO Q. Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem[J]. Nonlinear Analysis:Theory, Methods&Applications, 2005, 63(2):237-246.
[15] ROGERS C, SHADWICK W F. B(a|¨)cklund Transformations and Their Applications[M]. New York:Academic Press,1982.