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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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