New explicit solutions of the nonlinear wave equations with damping term

For the nonlinear wave equation with damping term introduced in [Z.J. Yang, G.W. Chen, Initial value problem for a nonlinear wave equation with damping term, Acta Math. Appl Sin. 23 (2000) 45 (in Chinese)]utt-2buxxt+auxxxx=β(uxn)x,α>0,β≠0,n⩾2,a new similarity reduction is presented by means of the direct reduction method due to Clark and Kruskal and the solutions for the equations with n⩾3 are given by using the new similarity reduction. By using similarity reductions and the ingenious construction of a proper auxiliary equation, some new special solutions such as one real solitary wave solution and one singular solitary wave solutions for β>0, and two complex solutions for β<0 are presented, respectively.