New type of exact solutions of nonlinear evolution equations via the new Sine–Poisson equation expansion method

In this paper, based on the well-known Sine–Poisson equation, a new Sine–Poisson equation expansion method with constant coefficients or variable coefficients is presented, which can be used to construct more new exact solutions of nonlinear evolution equations in mathematical physics. The KdV–mKdV equation and the typical breaking soliton equation are chosen to illustrate our method such that many types of new exact solutions are obtained, which include exponential solutions, kink-shaped solutions, singular solutions and soliton-like solutions.