## Explicit solitary-wave solutions to generalized Pochhammer-Chree equations.(English)Zbl 0935.35132

Summary: For the solitary-wave solution $$u(\xi) = u(x-vt+\xi_0)$$ to the generalized Pochhammer-Chree equation ($$PC$$ equation) $u_{tt}-u_{ttxx}+ru_{xxt}-(a_1u+a_2u^2+a_3u^3)_{xx}=0,\tag{1}$ $$r,a_i=\text{consts }(r\neq 0),$$ the formula $\int^{+\infty}_{-\infty} [u'(\xi)]^2 d\xi=\frac{1}{12rv}(C_+-C_-)^3 [3a_3(C_++C_-)+2a_2],\quad C_{\pm}=\lim_{\xi\to\pm\infty} u(\xi),$ is established. It is shown that the generalized $$PC$$ equation (1) does not have bell profile solitary-wave solutions but may have kink profile solitary-wave solutions. However a special generalized $$PC$$ equation $u_{tt}-u_{ttxx}-(a_1u+a_2u^2+a_3u^3)_{xx}=0,$ 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 $u_{tt}-u_{ttxx}-(a_1u+a_3u^3+a_5u^5)_{xx}= 0$ are proposed.

### MSC:

 35Q51 Soliton equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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### References:

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