广义Pochhammer-Chree方程的显式精确孤波解  

Explicit Solitary_Wave Solutions to Generalized Pochhammer_Chree Equations

作　　者：张卫国[1,2] 马文秀[1,2] 

机构地区：[1]长沙铁道学院数理力学系 [2]复旦大学数学系

出　　处：《应用数学和力学》1999年第6期625-632,共8页Applied Mathematics and Mechanics

摘　　要：首先对广义Ｐｏｃｈｈａｍｍｅｒ＿Ｃｈｒｅ方程（ＰＣ方程）ｕｔ－ｕｔｘｘ＋ｒｕｘｘｔ－（ａ１ｕ＋ａ２ｕ２＋ａ３ｕ３）ｘｘ＝０（ｒ≠０）（Ⅰ）的孤波解ｕ（ξ）建立了公式∫＋∞－∞［ｕ′（ξ）］２ｄξ＝１１２ｒｖ（Ｃ＋－Ｃ－）３［３ａ３（Ｃ＋＋Ｃ－）＋２ａ２］·由此推知：广义ＰＣ方程（Ⅰ）不可能有钟状孤波解，只可能有扭状孤波解；而广义ＰＣ方程ｕｔ－ｕｔｘｘ－（ａ１ｕ＋ａ２ｕ２＋ａ３ｕ３）ｘｘ＝０（Ⅱ）可能既有钟状孤波解又有渐近值满足３ａ３（Ｃ＋＋Ｃ－）＋２ａ２＝０的扭状孤波解·进一步求出了广义ＰＣ方程（Ⅰ）的扭状孤波解。For solitary_wave solutions u(ξ)=u(x-vt+ξ 0) to the generalized Pochhammer_Chree equation (PC equation) u tt -u ttxx +ru xxt -(a 1u+a 2u 2+a 3u 3) xx =0 r,a i= consts (r≠0),(Ⅰ) the formula ∫ +∞ -∞ 2 d ξ=112rv(C +-C -) 3[3a 3(C ++C -)+2a 2], C ±= lim ξ→±∞u(ξ), is established, by which it is shown that the generalized PC equations (Ⅰ) has not bell profile solitary_wave solutions but may have kink profile solitary_wave solutions. However a special generalized PC equation u tt -u ttxx -(a 1u+a 2u 2+a 3u 3) xx =0, a i= consts (Ⅱ) may have not only bell profile solitary_wave solutions, but also kink profile solitary wave solutions whose asymptotic values satisfy 3a 3(C ++C -)+2a 2=0. Furthermore all expected solitary_wave solutions are given. Finally some explicit bell profile solitary_wave solutions to another generalized PC equation u tt -u ttxx -(a 1u+a 3u 3+a 5u 5) xx =0, a i= consts (Ⅲ) are proposed.

关 键 词：非线性发展方程 广义 孤波解 精确解 P-C方程 

分 类 号：O175.26[理学—数学]