The traveling wave solutions of the perturbed double Sine-Gordon equation

In the paper, when \(0<c<1\) $0<c<1$ , the phase portraits, traveling wave solutions and the minimum positive period of the periodic orbit for the double Sine-Gordon equation are discussed by using the dynamical system method and variable transformation. With the help of Melnikov’s function, the geometric singular perturbation theory and symbolic computation, we prove that the existence of traveling wave solution of the perturbed double Sine-Gordon equation for \(0<c<1\) $0<c<1$ . The wave speed \(c=c(\alpha , \beta , \epsilon )\) $c=c(\alpha ,\beta ,ϵ)$ and the constraints of parameters \(\alpha \) $\alpha$ , \(\beta \) $\beta$ are given.