Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

Abstract

For the solitary-wave solution u(ξ) = u(x − vt + ¾0) to the generalized Pochhammer-Chree equation (PC equation) $$u_{tt} - u_{ttxx} + ru_{xxt} - (a_1 u + a_2 u^2 + a_3 u^3 )_{xx} = 0, r,a_i = consts(r \ne 0)$$ (I) the formula\(\int_{ - \infty }^{ + \infty } {[u'(\xi )]^2 d\xi = } \frac{1}{{12r\nu }}(C_ + - C_ - )^3 [3a_3 (C_ + - C_ - ) + 2a_2 ], C_ \pm = \mathop {\lim }\limits_{\xi \to \pm \infty } u(\xi )\), is established, by which it is shown that the generalized PC equation (I) does not have bell profile solitary-wave solutions but may have kink profile solitary-wave solutions. However a special generalized PC equation $$u_{tt} - u_{ttxx} - (a_1 u + a_2 u^2 + a_3 u^3 )_{xx} = 0, a_i = consts$$ (II) may have not only bell profile solitary-wave solutions, but also kink profile solitary-wave solutions whose asymptotic values satisfy 3a3(C++C−)+2a2=0. Furthermore all expected solitary-wave solutions are given. Finally some explicit bell profile solitary-wave solutions to another generalized PC equation $$u_{tt} - u_{ttxx} - (a_1 u + a_3 u^3 + a_5 u^5 )_{xx} = 0, a_i = consts$$ (III) are proposed.