Topological and singular soliton solution to Kundu-Eckhaus equation with extended Kudryashov's method

In this paper, we apply the extended Kudryashov method to a nonlinear Schrödinger type equation called the Kundu-Eckhaus equation or the Eckhaus equation which was independently introduced by Wiktor Eckhaus and by Anjan Kundu in 1984-1985 to model the propagation of waves in dispersive media. The proposed method is direct, effective and takes full advantages of the Bernoulli and Riccati equations to construct new exact solutions of that model and can be extended to many nonlinear evolution equations in mathematical physics.