广义Pochhammer-Chree方程的显式精确孤波解

张卫国; 马文秀

广义Pochhammer-Chree方程的显式精确孤波解

张卫国¹,
马文秀²

1.
长沙铁道学院数理力学系, 长沙 410075;
2.
复旦大学数学系, 上海 200433

详细信息

作者简介:
张卫国(1957- ),男,教授,博士.

中图分类号: O175.2
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- 被引次数: 0
出版历程
- 收稿日期: 1977-10-20
- 修回日期: 1999-02-04
- 刊出日期: 1999-06-15

Explicit Solitary-Wave Solutions to Generalized Pochhammer-Chree Equations

Zhang Weiguo¹,
Ma Wenxiu²

1.
Department of Mathematics and Mechanics, Changsha Railway University, Changsha 410075, P R China;
2.
Institute of Mathematics, Fudan University, Shanghai 200433, P R China;

摘要

摘要: 首先对广义Pochhammer-Chre方程(PC方程)u_tt-u_ttxx+ru_xxt-(a₁u+a₂u²+a₃u³)_xx=0(r≠0)(Ⅰ)的孤波解u(ξ)建立了公式∫_-∞^+∞[u'(ξ)]²dξ=1/12rv(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u_tt-u_ttxx-(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,求出了广义PC方程(Ⅱ)的钟状孤波解和渐近值满足2a₃(C₊+C_-)+2a₂=0的扭状孤波解。最后给出了广义PC方程u_tt-u_ttxx-(a₁u+a₃u³+a₅u⁵)_xx=0(Ⅲ)的显式孤波解。
- 非线性发展方程 /
- 广义Pochhammer-Chree方程 /
- 孤波解 /
- 精确解
Abstract: For the solitary-wave solution u(ξ)=u(x-vt+ξ₀)to the generalized Pochhammer-Chree equation(PC equation) u_tt-u_ttxx+ru_xxt-(a₁u+a₂u²+a₃u³)_xx=0,r,a_i=consts(r≠0),(Ⅰ). the formula ∫_-∞^+∞[u'(ξ)]²dξ=1/12rv(C₊-C_-)³[3a₃(C₊+C_-)+2a₂], 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-(a₁u+a₂u²+a₃u³)_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 3a₃(C₊+C_-)+2a₂=0. Furthermore all expected solitary-wave so-lutions are given. Finally some explicit bell profile solitary-wave solutions to another generalized PC equation u_tt-u_ttxx-(a₁u+a₃u³+a₅u⁵)_xx=0 a_i=consts are proposed.
- nonlinear evolution equation /
- generalized Pochhammer-Chree equation /
- solitary-wave solution /
- exact solution

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参考文献(3)

[1]	Bogolubsky I L.Some examples of inelastic so liton interaction[J].Computer Physics Communications,1977,13(2):149~155.
[2]	Clarkson P A,Le Veque R J,Saxton R.Solitary-wave interactions in elastic rods[J].Studies in Applied Mathematics,1986,75(1):95~122.
[3]	Saxton R.Existence of solutions for a finite nonlinearly hyperelastic rod[J].J Math Anal Appl,1985,105(1):59~75.