The modified exp($−\Omega(\zeta)$)-expansion function method is applied to the Wu–Zhang system with conformable time-fractional derivative to construct new analytical solutions. We have obtained some soliton-type solutions such as dark, singular and combined soliton solutions. We have seen that all solutions have provided the mentioned equation system. For the suitable value of the solutions, the 2D–3D and contour surfaces have been plotted.

This study aims to obtain travelling wave solutions of the doubly dispersive equation in nonlinear dynamic elasticity by the sine-Gordon expansion method. We give physical explanation of the presented solutions under suitable parameters via the 3D, 2D and contour simulations.