Combined soliton solutions of a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation in the cubic–quintic nonlinear material

We concentrate on a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation with cubic–quintic nonlinearities, and get some exact combined soliton solutions and periodic solutions based on the Jacobian elliptic function method and double function method. We discuss dynamical behaviors of bright soliton, dark soliton, periodic bright soliton train and periodic M-shaped soliton train, and find that shift distances along the x -direction of these solitons all add if the value of the fractional order increases.