New closed form soliton and other solutions of the Kundu-Eckhaus equation via the extended sinh-Gordon equation expansion method

In this paper, the extended sinh-Gordon equation expansion method (ShGEEM) is applied to look for new complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark-bright, singular, combined singular soliton and other soliton solutions are obtained of the Kundu-Eckhaus (KE) equation which demonstrates the physical explanation in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. By choosing randomly the special values of free parameters, the solitary wave solutions are shown different patterns. The reported results might play an important role in the quantum field theory and nonlinear optics for explaining the physical meaning of the studied model. The obtained results also demonstrate that the extended ShGEEM is more competent than the other applied method applied by authors in the literature. Therefore, it is shown that the applied method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations arises in mathematical physics and nonlinear optics.