广义Pochhammer-Chree方程的显式精确孤波解

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_-)33a₃(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。

Explicit Solitary_Wave Solutions to Generalized Pochhammer_Chree Equations

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_-)33a₃(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.